Non-commutative manifolds, the free square root and symmetric functions in two non-commuting variables

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in $\m^2$ we construct a 2-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables.


Introduction
Free analysis, the study of holomorphic functions in several non-commuting variables, dates back to 1973 and the seminal paper [22] by J. L. Taylor. The theory has picked up ever greater momentum in the past decade. The monograph [17], by D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov contains a panoramic survey of the field up to the time of its writing. Since then, there have been further breakthroughs in both geometry (see e.g. [9,12,13,14,15] and function theory (see e.g. [7,8,16]).
Taylor's founding idea for non-commutative analysis was that analytic functions in several non-commuting variables should have the same basic algebraic properties as free polynomials have when viewed as functions on tuples of matrices (of indeterminate size). Here a free polynomial means a polynomial in non-commuting variables over the field C of complex numbers.
Traditionally, free analysis has dealt with functions defined on subsets of the d-dimensional non-commutative universe M d , which comprises the space of d-tuples of square matrices, for d ≥ 1. Thus where M n denotes the algebra of n × n matrices over C and M d n = {(x 1 , x 2 , . . . , x d ) | x r ∈ M n for r = 1, 2, . . . , d}.
In this paper we introduce the notion of an nc-or non-commutative manifold, which bears the same relation to M d as complex manifolds bear to C d . This natural extension is needed even for such a basic notion as the "free Riemann surface" of the matricial square root function, an object that we construct in Section 6.
We were led to introduce topological nc-manifolds while seeking a noncommutative version of the anciently-known fact that any symmetric holomorphic function of d variables can be expressed as a holomorphic function of the elementary symmetric functions in d variables on a suitable domain.
In particular, if f (z, w) is a holomorphic function on C 2 that is symmetric, in the sense that f (z, w) = f (w, z) for all z, w ∈ C, then there exists a holomorphic function g on C 2 such that f (z, w) = g(z + w, zw) for all z, w ∈ C. We asked whether there is a non-commutative version of this result.
Any non-commutative analog must be consistent with results of M. Wolf [25], who studied an algebraic version of this question in 1936. She proved that the algebra of symmetric free polynomials in d non-commuting variables is not finitely generated when d > 1, and is in fact isomorphic to the algebra of free polynomials in countably many variables. It follows that there is no polynomial map π : M 2 → M d , for any d ∈ N, with the property that, for every symmetric free polynomial ϕ in 2 variables, there exists a free polynomial Φ such that ϕ = Φ • π; otherwise, the components of π would constitute a finite basis for the algebra of symmetric free polynomials. Nevertheless, we show that there is a close parallel to the classical result in the context of free analysis on nc-manifolds. The main result of the paper is Theorem 10.1.
Before stating a somewhat special case of the theorem, let us informally describe some key notions (precise definitions are in Sections 2 and 4). A subset of M d is called an nc set if it is closed with respect to direct sums. A function f defined on some subset of M d is called graded if f (x) ∈ M n whenever x ∈ M d n . A graded function on an nc set is called an nc function if it preserves direct sums and joint similarities. A graded function f on an arbitrary set D ⊆ M d is called conditionally nc if it preserves joint similarities, and, in addition, there is a graded functionf such that whenever x and x ⊕ y are in D, then f (x ⊕ y) = f (x) ⊕f (y). Theorem 1.2. Let S be a symmetric nc set in M 2 that is open in the free topology. There is subset S oo of S and a polynomial map π : M 2 → M 3 such that G oo = π(S oo ) is a topological nc-manifold in the Zariski-free topology, and such that there is a canonical isomorphism between bounded symmetric nc functions defined on S and bounded holomorphic functions on the manifold G oo that are conditionally nc.
The map π is given by π(x 1 , x 2 ) = ( 1 2 (x 1 + x 2 ), 1 4 (x 1 − x 2 ) 2 , 1 8 (x 1 − x 2 )(x 1 + x 2 )(x 1 − x 2 )). (1.4) It is generically 2-to-1 on M 2 , but there is a singular set on which it is many to one. This set is excluded in S oo , and both S oo and G oo can be given the structure of topological nc-manifolds with respect to a topology called the Zariski-free topology, defined in Section 7. There is also an isomorphism between Zariski-freely holomorphic symmetric functions f oo on S oo and Zariskifreely holomorphic functions on G oo (with no assumption of boundedness needed)-this is Theorem 9.15. The bulk of the paper comprises the establishment of a suitable notion of topological nc-manifold and construction of the manifolds G oo . Topological nc-manifolds are defined in Section 4. The nc-universe M d defined by equation (1.1) is unlike R d and C d in that it admits numerous natural topologiesa fact which adds an extra richness to the theory of topological nc-manifolds.
Section 5 describes the basic theory of free holomorphic functions of a single variable. In one variable such functions are determined by their action on scalars, and are given by the holomorphic functional calculus for matrices (Proposition 5.13).
In Section 6 we present a simple example of a one-dimensional free manifold -a non-commutative version of the Riemann surface of the square root function. This theory is an essential component of the construction of the manifold G.
In Section 7 we define the Zariski-free topology, which is rather subtle, but seems necessary in order to avoid certain singularities in the map π. In Section 8 we construct a Zariski-free nc-manifold G, which is the manifold which would be G oo in Theorem 1.2 if S were all of M 2 . Of course bounded symmetric functions are not of interest when S = M 2 , but there is a version of the theorem that does not require boundedness. This is Theorem 8.35. Theorem 1.5. There is a canonical bijection between (i) symmetric nc functions f that are freely holomorphic on M 2 , and (ii) holomorphic functions F oo defined on the Zariski-free manifold G that are conditionally nc and have the property that for every w ∈ M 2 , there is a free neighborhood U of w such that F oo is bounded on π(U ) ∩ G.
In Sections 9 and 10 we prove the main results of the paper, Theorems 10.1, 10.11 and 9.15. Finally, in Section 11, we give a non-commutative version of the classical Newton-Girard formulae for power sums in terms of elementary symmetric functions. There is a significant difference in the non-commutative context: there is no longer a finite algebraic basis for the algebra of symmetric polynomials. We can, however, derive explicit iterative formulae for writing the symmetric sums where n ∈ Z, as rational functions composed with the map π of equation (1.4).
2 Mappings and topologies on M d

Nc-functions
In this section we describe the basic objects of free analysis. Firstly, the ncsets in the nc-universe M d of equation (1.1) are the sets that are closed under direct sums. They are are the natural domains of definition of nc-functions, which are the M 1 -valued functions that are graded, preserve direct sums and respect similarity transformations. We now explain the precise meaning of these terms.
Let N denote the set of positive integers and, for n ∈ N, let I n = {M ∈ M n | M is invertible}.
Definition 2.1. If D ⊆ M d we say that D is an nc-set if D is closed with respect to the formation of direct sums, that is: For all n 1 , n 2 ∈ N and all x 1 ∈ D ∩ M d n 1 , x 2 ∈ D ∩ M d n 2 , 2. An nc-function in d variables is a function f whose domain is an nc-set D ⊆ M d , whose codomain is M 1 , and which satisfies A function f that has property (1) is said to be graded. for all x ∈ D 1 . If, in addition, F is a bijection and both F and F −1 are nc-mappings then we say that F is an nc-isomorphism.
J. Taylor proved [21] that properties (2) and (3) of Definition 2.2 are equivalent to the single property that f respects intertwinings, in the following sense: (4) for all n 1 , n 2 ∈ N, for x 1 ∈ D∩M d n 1 , for x 2 ∈ D∩M d n 2 and all n 2 -by-n 1 matrices L satisfying Lx j 1 = x j 2 L for j = 1, . . . , d, Lf (x 1 ) = f (x 2 )L. (2.4) 2.2 Topologies on the nc-universe Definition 2.5. Let τ be a topology on M d . We say that τ is an nc topology or equivalently an admissible topology if τ has a basis consisting of finitely open nc-sets.
As we have mentioned, the nc-universe M d admits several natural topologies [4]. In this paper we shall consider nc-manifolds based on M d endowed with three different topologies, the fine, free and Zariski-free topologies. The fine topology is the topology generated by all finitely open nc-sets. Domains do not need to be nc-sets. For example the set is a domain (see Proposition 6.5 below), indeed a free domain (see Definition 2.8), but is not an nc-set. We now describe the free topology. It was introduced in [1] in the context of a non-commutative Oka-Weil approximation theorem. For Observe that if δ 1 and δ 2 are matrices of free polynomials, then As a consequence of this fact, the collection of sets of the form B δ is closed with respect to finite intersections and thus forms a basis for a topology.
h is constant on the intervals [0, 1 2 ) and ( 1 2 , 1], and so freely continuous on those intervals. h is also freely continuous at the point 1 2 . For consider any basic free neighborhood B γ of the point h( 1 2 ) = x ⊕ y in U . Then we have γ(x ⊕ y) < 1, and so γ(x) < 1 and γ(y) < 1, which is to say that h(t) ∈ B γ for all t in the neighborhood [0, 1] of 1 2 . Thus h is continuous at 1 2 , and so h is a continuous path in U such that h(0) = x and h(1) = y. Proof. Path-connectedness implies connectedness. To say that M d is locally connected means that every point in M d has a neighborhood base of connected sets. The sets B δ comprise such a base.
Remark 2.11. The proof of Proposition 2.9 shows slightly more. For any freely open set U and any points x, y ∈ U such that x ⊕ y ∈ U , there is a freely continuous path in U from x to y.

Holomorphy with respect to admissible topologies
The notion of holomorphy for a function on M d depends on the chosen topology of M d . (2) f is τ -locally nc, that is, for each x ∈ D there exists an nc set U ∈ τ such that x ∈ U ⊆ D and f |U is an nc-function; (3) f is τ -locally bounded, that is, for each x ∈ D, there exists a τneighborhood U of x such that f |U is bounded.
At first sight this is a surprising definition. It is at least partially justified by its relation to the following notion of analyticity.  Proof. Consider a point x ∈ D∩M d n . Since ϕ is τ -holomorphic we may choose a τ -neighborhood U of x in D on which ϕ is an nc-function and another τneighborhood V on which ϕ is bounded. Since the topology τ is admissible, it is an nc-topology and so we may assume that V is an nc-set. Then U ∩ V is an nc-set and is a τ -neighborhood of x in D on which ϕ is a bounded nc-function. Theorem 4.10 of [1] asserts that under these hypotheses ϕ is analytic on U ∩ V ∩ M d n . Thus ϕ is analytic on some neighborhood of an arbitrary point of D ∩ M d n , and therefore ϕ is analytic on D ∩ M d n . Proof. Consider z, w ∈ D. Since z⊕w ∈ D, f (z⊕w) is defined. By Definition 3.1(2) there is an nc set U ∈ τ such that z ⊕ w ∈ U ⊆ D and f |U is an nc function. It follows that The following statement is routine to check.
Proposition 3.5. Let τ be an admissible topology on M d and let D be a τ -open set. The set of τ -holomorphic functions on D is an algebra with respect to pointwise operations. The set H ∞ τ (D) of bounded τ -holomorphic functions on D is a Banach algebra with respect to pointwise operations and the supremum norm The case where τ is the free topology will be of special interest here, and in this case we refer to τ -holomorphic functions as free (or freely) holomorphic functions. Such functions are particularly well behaved on account of the following theorem [1] (it is also proved in [6] and [3]).
is a free holomorphic function if and only if f can be locally uniformly approximated by free polynomials. That is, f is freely holomorphic if and only if for each x ∈ D there exists a free domain U satisfying x ∈ U ⊆ D with the property that for each ε > 0 there exists a free polynomial p such that sup y∈U f (y) − p(y) < ε.
To prove that freely holomorphic functions are freely continuous we need the following simple observation. In the Lemma and elsewhere the norm · on M d n is given by Lemma 3.7. A free polynomial is freely locally Lipschitz. That is, if p is a free polynomial in d variables and z ∈ M d then there exist a free neighborhood G of z and a positive constant K such that Proof. It is enough to prove the statement in the case that p is the monomial . . x r k for some k ∈ N and r 1 , . . . , r k ∈ {1, . . . , d}. Let Then x ∈ B δ if and only if x j < 1 + z for j = 1, . . . , d. Thus z ∈ B δ . We have, for any x, y ∈ M d , Hence, for every x, y ∈ B δ , Thus p is freely locally Lipschitz.
is freely holomorphic if and only if f is a freely locally nc-function (as in Definition 3.1 (2)) and f is continuous when U and M 1 are equipped with the free topologies.
Proof. First assume that U is an nc-set, f is an nc-function and f is continuous when D and M 1 are equipped with the free topologies. In the light of Definition 2.3 it suffices to show that f is locally bounded. Fix z ∈ U . As f is assumed continuous and U is open in the free topology, there exists a free matricial polynomial δ such that z ∈ B δ ⊆ U and such that This proves that f is locally bounded. Now assume that U is a free domain in M d and f is a free holomorphic function on U . The continuity of f will follow if we can show that f −1 (B q ) is freely open in M d whenever q is a square matrix of polynomials in one variable.
Fix z ∈ f −1 (B q ). By Theorem 3.6 there exists a square matrix δ of free polynomials such that f is bounded on B δ and a sequence of free polynomials p 1 , p 2 , . . . such that z ∈ B δ ⊆ U and It follows that the polynomials p k are uniformly bounded on B δ . By Lemma 3.7 q is locally Lipschitz on U , and hence Fix a strictly decreasing sequence t 1 , t 2 , . . . with t k > 1 for all k and t k → 1 as k → ∞. Use the property (3.10) to construct inductively a sequence k 1 , k 2 , . . . of positive integers such that

The last union in this formula is a union of basic freely open sets and so is a freely open set in
for all x ∈ D.
Let D 1 , D 2 be τ -open sets in M d . We say that F : D 1 → D 2 is a τ -biholomorphic mapping if F is a bijection and both F and F −1 are τholomorphic mappings. (1) F is a freely holomorphic map; (2) F is freely locally bounded; (3) F is freely continuous.
In particular, F is a freely biholomorphic map if and only if F is a homeomorphism in the free topology.
This result is proved in much the same way as Proposition 3.8.

Nc-manifolds and topological nc-manifolds 4.1 Nc-manifolds
In this section we define nc-manifolds, topological nc-manifolds and free manifolds. The concept of an nc-manifold is the generalization of the concept of an nc-set as defined in Definition 2.1 and as such is purely algebraic in nature. The notions of charts, atlases and transition functions transfer directly from the classical theory, only in the new context a chart is a bijective map from a subset of an nc-manifold to a subset of M d ; in the case of a topological nc-manifold the image of a chart is an open set in M d with respect to some specified topology.
Definition 4.1. If X is a set, then we say that α is a d-dimensional coordinate patch or chart on X if α is a bijection from a set U α ⊆ X to a set D α ⊆ M d . If α and β are a pair of d-dimensional co-ordinate patches on X with U α ∩ U β = ∅, then we define the transition map T αβ by If X is a set then we say that A is a d-dimensional nc-atlas for X if A is a collection of d-dimensional co-ordinate patches on X, and, for all α, β ∈ A, is a union of nc-sets, and (2) for every nc-subset W of α(U α ∩ U β ), the restriction of T αβ to W is an nc-mapping.
A d-dimensional nc-manifold is an ordered pair (X, A) where X is a set and A is a d-dimensional nc-atlas for X.
If (X 1 , A 1 ) and (X 2 , A 2 ) are nc-manifolds of dimensions d 1 , d 2 respectively, then a map f : X 1 → X 2 is an nc-map (or nc-mapping) if, for every α ∈ A 1 (with domain U α and codomain D α ) and β ∈ A 2 (with domain V β and codomain E β ) is a union of nc-sets, and is an nc-map.
If (X 1 , A 1 ) and (X 2 , A 2 ) are nc-manifolds then an nc-isomorphism from X 1 to X 2 is a bijective map f : X 1 → X 2 such that both f and f −1 are nc-maps.
Remark 4.5. 1. If (X, A) is an nc-manifold then, for every α ∈ A, D α is a union of nc-sets. This is a consequence of condition (1) in the definition of an nc-atlas above and the identity 2. If (X, A) is an nc-manifold then there is an nc-atlas A * on X such that the identity map id X is an nc-isomorphism from (X, A) to (X, A * ) and, for every α ∈ A * , the range of α is an nc-set (not merely a union of nc-sets). Indeed, we may define A * to be the set of all maps α|V for some α ∈ A and V ⊂ U α such that α(V ) is an nc-set. It would be possible to develop the theory of nc-manifolds with the assumption that the ranges of charts are always nc-sets, but in the topological context it is convenient to allow them to be merely unions of nc-sets.
To prove that id X : (X, A) → (X, A * ) is an nc-isomorphism, consider any α ∈ A and γ ∈ A * , say γ = β|V where V ⊂ U β and β(V ) is an nc-subset of is an nc-set and T βα is an nc-map, it follows that W αγ is an nc-set. Moreover, for is an nc-map. Similarly, reversing the roles of A and A * , we obtain, for any is also an nc-map. Thus id X is an nc-isomorphism with respect to A and A * .

Topological nc-manifolds
We now consider the case where X, in addition to carrying the structure of an nc-manifold, is a topological space. X will be a topological nc-manifold if it is locally homeomorphic to an open set in some M d . Since there is no one "correct" topology to place on M d , we first fix a topology τ on M d . We assume that τ is an admissible topology in the sense of Definition 2.5.
If X is a set and T is a topology on X, then we say that α is a topological d-dimensional nc-co-ordinate patch on (X, T ) with respect to τ if α is a ddimensional nc-co-ordinate patch on X and, in addition, Here U α ⊆ X is a T -open set equipped with the relative topology induced by T and D α ⊆ M d is a τ -open nc-set equipped with the relative topology induced by τ . Since τ is locally nc, D α is a union of open nc-sets. If (X, T ) is a topological space then we say that A is a d-dimensional topological nc-atlas for (X, T ) with respect to τ if A is a collection of topological d-dimensional nc-co-ordinate patches on (X, T ) and A is an nc-atlas for X with respect to τ in the sense of Definition 4.1.
Definition 4.7. Let τ be an admissible topology on M d . A topological ddimensional nc-manifold with respect to τ is an ordered triple (X, T , A) where X is a set, T is a topology on X and A is a d-dimensional topological nc-atlas for (X, T ) with respect to τ .
In the special case where τ is the free topology we say that X is a free manifold.
The manifolds studied in analysis usually have some smoothness property, such as C ∞ or analyticity, whereas the topological nc-manifolds introduced in Definition 4.7 are not assumed smooth. It is simple to extend the notion of topological nc-manifold further to bring in appropriate notions of smoothness.
Definition 4.8. Let τ be an admissible topology on M d . A d-dimensional holomorphic nc-manifold with respect to τ is a topological d-dimensional ncmanifold (X, T , A) with respect to τ such that, for all α, β ∈ A, the transition map T αβ is a τ -holomorphic mapping in the sense of Definitions 3.11 and 3.1.
If (X, T , A) is a d-dimensional holomorphic nc-manifold with respect to τ then a function F : X → M 1 is said to be holomorphic on X if, for every α ∈ A, the map More generally, we can define the notion of a holomorphic map between two topological nc-manifolds. Definition 4.9. Let τ 1 , τ 2 be admissible topologies on M d 1 , M d 2 respectively, and let (X j , T j , A j ) be a topological d j -dimensional manifold with respect to τ j for j = 1, 2. A map f : X 1 → X 2 is said to be holomorphic at a point x ∈ X 1 if there exist α ∈ A 1 (with domain U α and domain D α ) and β ∈ A 2 (with domain V β and range E β ) such that x ∈ U α , f (x) ∈ V β and the map Note that the definition is independent of τ 2 , save for the requirement that τ 2 be admissible.
In the case that τ is the free topology, Proposition 3.12 tells us that continuity implies holomorphy. Accordingly, free manifolds are a precise noncommutative analog of complex manifolds. Indeed, they are the topological manifolds that are equipped with an atlas of homeomorphisms onto free domains in M d with the property that the transition functions are freely biholomorphic maps. To see this fact, assume that (X, T , A) is a free manifold, α, β ∈ A, and U α , U β ∈ T with U α ∩ U β = ∅ . As U α ∩ U β ∈ T , the hypothesis (4.6) implies that α(U α ∩ U β ) and β(U α ∩ U β ) are open in the free topology. The hypothesis (4.6) also implies that is a homeomorphism. Hence, by Proposition 3.12, β •α −1 is a free biholomorphic mapping. Conversely, if the transition functions are assumed to be free biholomorphic mappings, then Proposition 3.12 implies that the transition functions are free holomorphic mappings.
If (X, T , A) is a topological d-dimensional nc-manifold with respect to τ then the transition function T αβ is a composition of two homeomorphisms and hence is a homeomorphism between α(U α ∩ U β ) and β(U α ∩ U β ) in their respective τ topologies, as well as being an nc-isomorphism between nc-sets when restricted to any nc-subset of α(U α ∩ U β ).
If (X, T , A) is a topological d-dimensional nc-manifold with respect to τ and τ is a finer admissible topology on M d then we define the topology T on X to be the one for which a base is Then (X, T , A) is a d-dimensional topological nc-manifold with respect to τ . The topology T is finer than T and the map id X : (X, T , A) → (X, T , A) is a holomorphic map of topological nc-manifolds.
In particular we may take τ to be the fine topology on M d . Then the topology T is the finest topology for which X is a topological nc-manifold.

Free holomorphic functions in one variable
In this section we show that a free holomorphic function in one variable is determined (via the functional calculus) by its restriction to M 1 1 . Let R + denote the set of positive real numbers. If c ∈ C k and r ∈ R + k , we define ∆(c, r) ⊆ C by where σ(x) denotes the spectrum of the matrix x. In the sequel we make the standing assumption that the radii r 1 , . . . , r k are so small that the discs c 1 + r 1 D, . . . , c k + r k D are pairwise disjoint.
Proof. Assume that M ∈ D(c, r). We construct a polynomial δ satisfying Choose a polynomial q satisfying This implies that q(σ(x)) ⊆ D. But then, which implies that x ∈ D(c, r).
If f is holomorphic on a neighborhood of ∆(c, r) and x ∈ D(c, r) then we may employ the Riesz Functional Calculus to define f ∧ (x) by the formula where s ∈ R + k is chosen so that σ(x) ⊆ ∆(c, s) and ∆(c, s) − ⊆ ∆(c, r).
Proof. It is straightforward to verify that f ∧ is an nc-function. The proposition will follow if we can show that f ∧ is locally bounded. To that end, fix , the proof of Proposition 5.3 will be complete if we can prove the following two claims.
To prove Claim 1, first notice that by Proposition 5.1, it suffices to show that G is a free domain. In the light of equation (5.6) and the fact that S is finite it will follow that G is a free domain if we can show that for each w ∈ C and each C > 0, But by Theorem 10.1 in [1] for each fixed w ∈ C, g w (x) = (w − x) −1 is a free holomorphic function on the free domain {x ∈ M 1 | w ∈ σ(x)}. As Proposition 3.8 guarantees that g w is freely continuous, it follows that is a free domain.
To prove Claim 2, let z ∈ [c, s] and let x ∈ G. As z ∈ [c, s], inequality (5.5) guarantees that we may choose w ∈ S such that But as inequalities (5.7) and (5.8) imply that This completes the proof of both Claim 2 and the theorem. Definition 5.9. For any free domain in U in M 1 we define U 1 to be U ∩ M 1 1 . If f is a free holomorphic function on U we define a function f 1 on U 1 by In the sequel we make no distinction between z ∈ C and [z] ∈ M 1 1 , and in particular, view U 1 both as a subset of C and as a subset of M 1 .
Proof. Suppose that M ∈ U and z ∈ σ(M ). We wish to show that z ∈ U 1 or equivalently, that z ∈ U .
Choose an I × J matrix of polynomials δ such that M ∈ B δ ⊆ U and choose a unit vector v ∈ C n such that M v = zv. We view δ(M ) as a linear transformation from C J ⊗ C n to C I ⊗ C n . As M ∈ B δ , there exists r < 1 such that δ(M ) ≤ r.
With the setup of the previous paragraph, if c ∈ C J , This proves that δ(z) ≤ r < 1. Hence z ∈ B δ ⊆ U , as was to be proved. Proof. Fix z 0 ∈ U 1 . As f is free holomorphic, by Theorem 3.6 f can be locally uniformly approximated by polynomials. Choose δ such that z 0 ∈ B δ ⊆ U and f can be uniformly approximated by polynomials on B δ . By continuity, there exists ε > 0 such that {z ∈ C | |z − z 0 | < ε} ⊆ B δ . It follows that f 1 can be uniformly approximated on a neighborhood in C by polynomials. Hence f 1 is holomorphic on a neighborhood of z 0 . As z 0 ∈ U 1 was chosen arbitrarily, it follows that f 1 is holomorphic on U 1 .
Proposition 5.13. If U ⊆ M 1 is a free domain and f is a free holomorphic function defined on U , then for all x ∈ U . Moreover f (x) belongs to the algebra generated by x and the identity matrix of appropriate type.
Proof. First observe that the assertion of the proposition makes sense. If x ∈ U , then Proposition 5.10 implies that σ(x) ⊆ U 1 . Also, if f is freely holomorphic on U , then Proposition 5.12 implies that f 1 is holomorphic on 2) for all x ∈ U . If p is a polynomial, then p ∧ 1 = p 1 and p 1 = p. Hence, the result holds in the special case where f = p. The general case then follows by approximation.

√ x
In this section we shall define √ x as a holomorphic function on a onedimensional free nc-manifold. By gluing together locally defined branches of the inverse of the free holomorphic function f (x) = x 2 we construct a free nc-manifold in much the same way that elementary textbooks construct the Riemann surface for √ z by piecing together locally-defined function elements. We obtain a locally finitely-sheeted one-dimensional free nc-manifold which has properties analogous to the Riemann surface for √ z. The zero matrix in M 2 has infinitely many square roots, but only one of them lies in the algebra generated by the zero and identity matrices. By Proposition 5.13, for any free holomorphic function f and any x for which f (x) is defined, f (x) lies in the algebra alg(x) generated by x and the identity. We shall therefore use the symbol √ x in the following way.
the n × n identity matrix, then We leave the proof of the following proposition to the reader.
Proposition 6.2. The free square root of a matrix x ∈ M 1 is empty if and only if the Jordan canonical form of x contains a nilpotent Jordan cell of type k × k for some k ≥ 2.
is not a freely open set: it contains 0 2×2 , but any basic free neighborhood of Ξ contains a matrix 0 t 0 0 for some t = 0. For the purpose of constructing a Riemann surface we consider the restriction of the square root function to Ξ o , the interior of Ξ in the free topology. We shall show (Proposition 6.6) that Ξ o is the set of nonsingular matrices, and that the union of the sets {x} × √ x, as x ranges over all nonsingular matrices, can be given the structure of a free nc-manifold. Let Since Hence I is open in the free topology. I is clearly closed under direct sums, and so it is freely connected, by Proposition 2.9. Now fix M ∈ Q. Let σ(M ) = {c 1 , . . . , c k } and choose r 1 , . . . , r k ∈ R + such that ∆(c, r) ∩ ∆(−c, r) = ∅. Let q be the characteristic polynomial of M and define δ = ρq where ρ ∈ R + is chosen so large that Hence Q is open in the free topology.
Consider any points x, y ∈ Q. Although Q is not closed under ⊕, since σ(x) can meet σ(−y), it is nearly so. Choose t ∈ (0, 1] such that σ(tx) is disjoint from σ(−y); then tx ⊕ y ∈ Q. Remark 2.11 shows that there is a freely continuous path in Q from tx to y, while there is an obvious freely continuous path in Q from x to tx. Proposition 6.6. The interior Ξ o of the set Ξ in the free topology is I.
Proof. By Proposition 6.5, I is freely open, and it is clearly contained in Ξ. Suppose I is a proper subset of Ξ o ; then there is a singular matrix M ∈ Ξ and a basic free neighborhood B δ of M contained in Ξ. Since B δ is invariant under unitary conjugations, we may assume that M is upper triangular, and since M is singular, we may take M to have zero as its (1, 1) entry. Let N = M ⊕ M ; then N ∈ B δ , and for a suitable permutation matrix P , the matrix P * N P in B δ is upper triangular and has 0 2×2 as a block in the (1, 1) position. For some complex ζ = 0, B δ contains the upper triangular matrix T differing from P * N P only in that its (1, 2) entry is ζ. If e 1 , e 2 , . . . denotes the standard basis of C n , for the appropriate n, then a Jordan chain for T corresponding to the eigenvalue 0 is e 2 , ζe 1 . Hence the Jordan form of T has a nilpotent Jordan cell of type at least 2 × 2, and therefore, by Proposition We shall construct the Riemann surface for √ x by piecing together function elements over I. Definition 6.7. By a free function element over I is meant a pair (f, U ) where U is a free domain in I and f is a free holomorphic function on U . We say a function element Proof. Let (f, U ) and (g, V ) be branches of √ x and assume that M ∈ U ∩ V and f (M ) = g(M ). By Proposition 5.13 there exist holomorphic functions Furthermore, since f and g are branches of √ x on U and V respectively, It follows from equation (6.9) that Since each component of ∆(c, r) meets σ(M ), it follows from equation (6.10) that f 1 = g 1 on ∆(c, r). Therefore, by Proposition 5.13, for all x ∈ D(c, r). Since Proposition 5.1 guarantees that D(c, r) is a free domain, the lemma follows by choice of The following definition expresses the Riemann surface for √ x as a union of graphs of function elements. This approach follows Chapter 8 of [5] quite closely. An alternative approach, based on cross-sections of a sheaf of germs of free holomorphic functions over I, is also possible. However, in the simple special case we are considering, this latter approach would amount to little more than a change in notation.
Lemma 6.12. There exists a unique topology T on R such that B is a basis for T .
This assertion follows immediately from Lemma 6.8.
Definition 6.13. Let R be equipped with the topology T of Lemma 6. 12.
Theorem 6.14. (R, T , A) is a free manifold.
Proof. The theorem follows from the following four facts each of which is a simple consequence of the previous constructions: According to Definition 6.11, if (f, U ) ∈ S then U is a free domain. So 2. holds.
To see 3., assume that V is a free domain in U . Then As in the classical case, the point of the Riemann surface for a multivalued function f on a domain D is that f can be regarded as a single-valued holomorphic function on the Riemann surface, which is a holomorphic manifold lying over D. The following statement makes this notion precise in the context of the matricial square root. Consider a point w ∈ R: then w ∈ graph(f, U ) for some function element (f, U ), which is to say that w = (x, f (x)) for some x ∈ U . We shall say that w lies over x.
Proof. Define F to be the restriction to R of the co-ordinate projection It is interesting to observe that, in contrast to the commutative case, where the Riemann surface for √ z lies over C \ {0} as a 2-sheeted surface, in the noncommutative case there is no bound on the number of sheets in R that lie over a given point M ∈ I. Proof. The proof uses the simple observation from linear algebra that a matrix M has exactly 2 k square roots that lie in alg(M ), the algebra generated by M . Fix M ∈ I and assume that |σ(M )| = k.
To see that there exist at least 2 k matrices N such that (M, N ) ∈ R, let σ = {c 1 , . . . , c k } and choose r ∈ R + k so that ∆(c, r) has k components and 0 ∈ ∆(c, r). Each of the 2 k distinct choices of square roots b 1 , . . . , b k for the points c 1 , . . . , c k gives rise to a distinct holomorphic branch In turn each of these distinct holomorphic branches of h b gives rise to a distinct function element Equally interesting is to observe that despite the phenomenon described in the preceding proposition, R is isomorphic to a free domain in M 1 .
Proposition 6.17. The map σ : Q → R defined by the formula Proof. Clearly, σ is injective and onto.
is a free holomorphic function defined on {x | x ∈ U }. Therefore σ is a free biholomorphic mapping.

The Zariski-free topology
We come now to a modification of the free topology on M d that will be needed for the construction of the topological nc-manifold G with properties described in Theorems 8.33 and 8.35.

Thin sets
Recall that a set T in a domain U ⊆ C d is said to be thin if at each point z ∈ U there exists an open neighborhood V of z in U and a nonconstant holomorphic function f on V such that f = 0 on V ∩ T . Simple facts are that thin sets are nowhere dense, closures of thin sets are thin sets and finite unions of thin sets are thin sets. A more subtle property of thin sets will be fundamental in later sections: if T is a thin set in U then every holomorphic function f on U \ T that is locally bounded on U has a unique holomorphic extension to all of U [20, Theorem I. 3.4]. Here the boundedness hypothesis is defined as follows.
It is false that every holomorphic locally bounded function on U \ T , where T is thin and U is a domain in C d , has a holomorphic extension to U . An easy counterexample is It is essential that f be locally bounded on U for a holomorphic extension to exist.

The Zariski-free topology
Let D be a free domain. By a free variety in D we mean a set V ⊆ D that has the form for some set S of freely holomorphic functions on D. Note that the zero symbol in equation (7.2) stands simultaneously for square zero matrices of all orders. To make matters precise we define where 0 n denotes the zero matrix in M n . Thus the definition (7.2) can equally be written We shall be loose about distinguishing 0 and 0. The following statement illustrates just one of the many surprises that result from the free topology not being Hausdorff. Free varieties in a free domain D are not necessarily relatively closed in D in the free topology.
Since p(x) is singular, there is an eigenvalue λ of x such that p(λ) = 0. Let u ∈ C n be a corresponding eigenvector. Consider any basic free neighborhood B δ of x in D, where δ is an m×m matrix of polynomials in one variable; then δ(x) ∈ M mn and δ(x) Mmn < 1. For any choice of ζ = (ζ 1 , . . . , ζ m ) ∈ C m , δ ij (x)ζ j u = δ ij (λ)ζ j u for i, j = 1, . . . , m.
Let ζ ⊗ u denote the nm × 1 matrix [ζ 1 u T . . . ζ m u T ] T , where the superscript T denotes transposition. The last equation shows that Take norms of both sides in C mn : Choose ζ to be a unit maximizing vector for δ(λ), so that Combining equations (7.4) and (7.5), we find that Since δ(x) < 1, and hence δ(λ) Mm < 1.
That is, λ ∈ G δ . Since p(λ) = 0, λ ∈ T . We have shown that every basic free neighborhood of x in D meets T , and so x is in the free closure of T . Conversely, suppose that x ∈ D ∩M n and p(x) is nonsingular. Then there is a basic free neighborhood B δ of x in M 1 that is disjoint from T . Indeed, we may choose δ to be the 1 × 1 polynomial c(g • p) where g is the characteristic polynomial of p(x) and c > 1/| det p(x)|. By the Cayley-Hamilton theorem, δ(x) = cg(p(x)) = 0, so that x ∈ B δ . For any m ∈ N and M ∈ T ∩ M m , δ(M ) = cg(p(M )) = cg(0 m ) = cg(0 1 )1 m = c(det p(x))1 m , and therefore Many of the important results about varieties in commutative analysis depend critically on the fact that varieties are relatively closed. Accordingly the following modification of the free topology is natural and fruitful in the nc context, as we shall see in Subsection 8.5.
is a finite union of free varieties in D.
In the definition, the set T is taken to be a finite union of free varieties. This is because, in contrast to the commutative case, a finite union of free varieties is not in general a free variety. The set-theoretic identity thus implies that the collection of basic Zariski-free sets in M d is closed with respect to finite intersections and hence forms a base for a topology.
Definition 7.11. The Zariski-free topology on M d is the topology that has as a base the collection of basic Zariski-free sets. A set that is open in the Zariski-free topology is a Zariski-free domain.
Note that there is also a smaller base for the Zariski-free topology, consisting of the sets B δ \ T where δ ranges over all matricial free polynomials and T ranges over all finite unions of free varieties in B δ . This base has the additional feature that it consists of nc-sets. Consequently the topology is an nc topology. It is clearly coarser than the finitely open topology. We deduce, in the terminology of Definition 2.5: We now explore some important relationships between free and Zariskifree domains and holomorphic functions. Proposition 7.14. If D 1 and D 2 are free domains and f : D 1 → D 2 is a free holomorphic mapping then f is continuous in the Zariski-free topology.
where D ⊆ D 2 is a free domain and T is a finite union of free varieties in D. Suppose that, as in Definition 7.8, = 0 for all g ∈ S i } and S i is a set of free holomorphic functions on D. Then is therefore Zariski-freely open. Proof. Let z ∈ D ∩ T . Choose w ∈ D \ T . Then z ⊕ w ∈ D \ T , and so Hence, by the intertwining property (2.4) of nc functions, Similarly, for each j, Hence ϕ(z ⊕ w) = a 0 0 ϕ(w) for all z ∈ D ∩ T and w ∈ D \ T . Extendφ to D by definingφ to agree with ϕ on D \ T . We claim thatφ is a freely holomorphic extension of ϕ from D \ T to D. Thatφ(z 1 ⊕ z 2 ) = ϕ(z 1 ) ⊕φ(z 2 ) is immediate if both z 1 and z 2 are in D \ T , since ϕ is an nc function by Proposition 3.4. It is also immediate if just one of z 1 , z 2 is in D \ T , by choice of the other as w. Assume therefore that both z 1 and z 2 are in D ∩ T , and choose any w ∈ D \ T . The relation is true by the definition ofφ if z 1 ⊕ z 2 ∈ T , and by the fact thatφ extends ϕ together with the nc property of ϕ if Comparison of equations (7.18) and (7.19) reveals that To see thatφ preserves similarity, consider x ∈ D and s ∈ I such that s −1 xs ∈ D. If x / ∈ T then also s −1 xs ∈ D \ T , and sô It remains to show thatφ is freely locally bounded, hence freely holomorphic. Let z be any point in D, and choose w ∈ D \ T . There is a Zariski-free basic neighborhood B δ \ S containing z ⊕ w on which ϕ is bounded, by M say. For every x ∈ B δ , is bounded by M , soφ is bounded on the free neighborhood B δ of z.
To see that the extension is unique, it is sufficient to prove that if ψ is a free nc holomorphic function on D that vanishes on D \ T , then it is identically zero. Suppose not. Then there is some point z ∈ T ∩ D such that ψ(z) = 0. There is some point w ∈ D \ T . Then z ⊕ w is in D \ T , but ψ(z ⊕ w) = ψ(z) ⊕ ψ(w) = 0, a contradiction. Lemma 7.20. Let ϕ : B δ \ T → B γ \ S be a Zariski-freely holomorphic nc map. Then ϕ extends to a unique freely holomorphic mapφ : B δ → B γ . Moreover, if ϕ is a Zariski-free homeomorphism, thenφ is a free homeomorphism.
Proof. Since B γ is an nc set, we may apply Lemma 7.15 to each component of ϕ in turn to obtain a freely holomorphic nc mapφ on B γ . We must show that the range ofφ is contained in B γ . This is automatic for points in B δ \ T , so consider a point x ∈ T ∩ B δ . Suppose γ(φ(x)) ≥ 1. Then there are unit vectors u, v such that | γ(φ(x))u, v | ≥ 1.
The uniqueness ofφ follows by Lemma 7.15.
Thusψ is the inverse ofφ.
where T α is a finite union of free varieties in B δα such that (i) whenever B δα ∩ B δ β is non-empty, so is (B δα \ T α ) ∩ (B δ β \ T β ); (ii) the function ϕ is bounded and nc on each set B δα \ T α for α ∈ A.
Then ϕ extends to a unique freely holomorphic functionφ on Proof. By Lemma 7.15, ϕ can be extended to a unique free function on each B δα . To see that these extensions coincide on each B δα ∩ B δ β , we observe that, by condition (i), if this intersection is non-empty, then there is a point w in (B δα \ T α ) ∩ (B δ β \ T β ). We can use this point w in equation (7.17) to construct the extension of ϕ to both B δα and B δ β , and so the extension will agree on the intersection.
The next result shows how close Zariski-free holomorphy is to free holomorphy. Proof. Let U be a free domain in M d and let ϕ : U → M 1 be a graded function. Suppose that ϕ is freely holomorphic -that is, it is freely locally nc and freely locally bounded on U . Since the Zariski-free topology is finer than the free topology, it is immediate that ϕ is Zariski-freely locally nc and Zariski-freely locally bounded. Thus ϕ is Zariski-freely holomorphic.
Conversely, suppose that ϕ is Zariski-freely holomorphic on U . Let x ∈ U . There is a Zariski-free open set D \ T containing x on which ϕ is nc and bounded. Therefore there exists B δ such that B δ \ T contains x and ϕ is nc and bounded on B δ \ T . By Lemma 7.15, there is a bounded nc extensionφ of ϕ| B δ \T to B δ . If y ∈ B δ ∩ T , then Soφ agrees with ϕ on all of B δ , including T . Therefore ϕ is freely locally bounded.

An nc-manifold for symmetrization
In this section we construct a two-dimensional topological nc-manifold G such that the algebra of holomorphic functions on G is canonically isomorphic (in a sense to be made precise in Theorem 8.35 below) to the algebra of symmetric free holomorphic functions in M 2 .
In the commutative case, if Ω ⊂ C 2 is a symmetric domain (that is, if (z, w) ∈ Ω implies that (w, z) ∈ Ω) then the domainΩ def = {(z + w, zw) : (z, w) ∈ Ω} has the property that the symmetric holomorphic functions f on Ω are in bijective correspondence with the holomorphic functions F onΩ via the relation f (z, w) = F (z + w, zw) for all (z, w) ∈ Ω.
We shall construct a topological nc-manifold G with properties analogous to those ofΩ for the symmetric free domain Ω = M 2 .
In [2] the algebra of symmetric free holomorphic functions on the biball B 2 , the noncommutative analog of the bidisc, is analysed by means of operatortheoretic methods. The authors constructed a set Ω ⊂ M ∞ and a map π : B 2 → Ω such that every symmetric free holomorphic function ϕ on B 2 can be expressed as Φ • π for some holomorphic nc-function Φ on Ω.
Here Ω is an infinite-dimensional set, but it can nevertheless be described in terms of only three noncommuting variables, provided that inverses and square roots are allowed. In this paper we adopt a more topological approach and obtain a 2-dimensional topological nc-manifold with properties analogous to Ω, but for symmetric functions on an arbitrary symmetric free domain. The theme that three variables suffice is reflected in the fact that the manifold we obtain in this section is presented as a subset of M 3 . However the topology of the manifold structure of G is not that induced by any natural topology of M 3 .

A geometric lemma
In this and the next two subsections we describe some simple combinatorial geometry and properties of free square roots, which play an essential part in the construction of G.
Throughout Section 8 the symbol ∆ will be reserved for sets in the plane that have the form Alternatively, if γ denotes the finite set {c 1 , c 2 , . . . c k }, We refer to such sets as simple sets with radius r. We define the separation of a simple set ∆ = γ + rD, denoted by sep ∆, by sep ∆ = min{|c − d| : c, d ∈ γ, c = d}. Definition 8.1. A simple set ∆ with radius r is t-isolated if r < t sep ∆. If ∆ 1 and ∆ 2 are simple sets, then ∆ 2 is subordinate to ∆ 1 if each disc in ∆ 2 meets at most one disc in ∆ 1 . Lemma 8.2. Let ∆ 1 and ∆ 2 be simple sets with radii r 1 and r 2 respectively. If ∆ 1 and ∆ 2 are 1 4 -isolated, then either ∆ 1 is subordinate to ∆ 2 or ∆ 2 is subordinate to ∆ 1 .
Proof. Suppose not, so that neither ∆ j is subordinate to the other. Since ∆ 1 is not subordinate to ∆ 2 there is a disc c + r 1 D in ∆ 1 that meets two discs d j + r 2 D, j = 1, 2 in ∆ 2 , with d 1 = d 2 . Pick points ζ 1 , ζ 2 such that Then |ζ j − c| < r 1 and |ζ j − d j | < r 2 .

Holomorphic square roots on the sets ∆ γ
Let Γ denote the set of finite subsets of C \ {0}. For each γ ∈ Γ we fix throughout the remainder of the section a simple set where r γ is chosen so that This choice of r γ guarantees that, for all γ ∈ Γ, Notice that the statement (8.5) implies that ∆ γ is a finite union of open discs whose closures are pairwise disjoint. For γ ∈ Γ let |γ| denote the cardinality of γ. Define 2 |γ| functions on ∆ γ in the following way. For each τ ∈ {−1, 1} γ , define ι γτ : ∆ γ → {−1, 1} by the formula ι γτ (z) = τ (c) if z ∈ c + r γ D.
The functions ι γτ are holomorphic on ∆ γ , indeed, they precisely consist of the 2 |γ| holomorphic square roots of the constant function 1 on ∆ γ . For each γ ∈ Γ, property (8.4) and the remark following statement (8.5) imply that there exists a branch of √ z defined on ∆ γ .
Convention 8.6. For every γ ∈ Γ, fix a holomorphic branch s γ of the square root function on ∆ γ .
Once this s γ is chosen, the other branches of √ z on ∆ γ can be described in terms of s γ with the aid of the functions ι γτ ; we define s γτ to be the holomorphic square root function s γ ι γτ on ∆ γ . Lemma 8.7. Assume γ ∈ Γ. A function f on ∆ γ is a holomorphic function satisfying f (z) 2 = z for all z ∈ ∆ γ if and only if there exists τ ∈ {−1, 1} γ such that f = s γτ .

Free square roots on the sets D γ
For each γ ∈ Γ define D γ ⊆ M 1 by With the help of the functions s γ and ι γτ we may define free holomorphic functions on D γ by means of the Riesz functional calculus. Since x ∈ D γ implies that σ(x) ⊆ ∆ γ , we may use the formula (5.2) to define, for every γ ∈ Γ, every τ ∈ {−1, 1} γ and every x ∈ D γ , Since s γ (z) 2 = z and ι γτ (z) 2 = 1 for z ∈ ∆ γ , S γτ (x) 2 = x and I γτ (x) 2 = 1 for all γ ∈ Γ and x ∈ D γ .
Notice that, by Proposition 5.3, S γτ and I γτ are free holomorphic functions on the free domain D γ . Moreover, every square root of x ∈ D γ in the algebra generated by x has the form S γτ (x) for some τ ∈ {−1, 1} γ .

The Zariski-free domain U γ
For each γ ∈ Γ we shall define a Zariski-free open set in M 2 having a certain genericity property. Let As D γ is a free domain, so is W γ . Furthermore, for fixed γ ∈ Γ and τ ∈ {−1, 1} γ , as I γτ is a free holomorphic function, defines a free holomorphic function on W γ . It follows that is a free variety in W γ .
In the constant cases, where either τ = 1 (that is, τ (c) = 1 for all c ∈ γ) or τ = −1 (that is, τ (c) = −1 for all c ∈ γ), we have either I γτ = 1 or I γτ = −1. Thus, in these two cases where τ is constant, V γτ is all of W γ . In the sequel, we express the condition that τ is not constant by writing τ = ±1.
Since, for each τ ∈ {−1, 1} γ , V γτ is a free variety in W γ , defines a set that is open in the Zariski-free topology of M 2 , though not, generally, in the free topology, since the varieties V γτ are typically not freely closed. We shall construct the Zariski-free manifold G by gluing together sheets in M 3 that lie over the domains U γ ⊂ M 2 . We can express the definition of U γ as follows: x ∈ D γ and u does not commute with I γτ (x) for any nonconstant τ ∈ {−1, 1} γ }. (8.14) The following statement is easy to see.

The definition of G
To motivate the ensuing definition of the set G we recount some ideas from [2]. It has long been known [25] that the algebra A of symmetric complex polynomials in two noncommuting variables w 1 and w 2 is not finitely generated. However, it was found in [2] that there is a close substitute for a finite basis. If we define then an algebraic basis of A is u, v 2 , vuv, vu 2 v, . . . , vu j v, . . . (8.16) Although this is an infinite basis of A, the relation shows that every symmetric free polynomial in w 1 , w 2 can be written as a rational expression in terms of the first three terms of the basis (8.16). Accordingly, it appeared that these three basic polynomials might have the potential to play the role that the elementary symmetric functions w 1 + w 2 and w 1 w 2 play in the scalar theory, though some extra complications result from the fact that rational expressions are required to represent polynomials. Thus, the underlying idea is to study the image of M 2 under the map (u, v 2 , vuv). Here, we realise this approach. We write x for the variable v 2 (so that v is a square root of x), and then invoke the structure theory for free holomorphic square roots developed in Section 6. Our strategy for the construction of a topological nc-manifold with the desired 'universal symmetrization' property will be to apply the maps (u, x, The expectation is that the many branches of the square root lead to coordinate patches on the image set that can be pieced together to yield the required topological nc-manifold. Our first attempt to execute this strategy and thereby construct a free nc-manifold failed because of singular behavior on certain subvarieties. The notion of the Zariski-free topology enabled us to circumvent this difficulty -see Remark 8.28 below. Let us first establish that the polynomials (8.16) do indeed constitute an algebraic basis for the algebra of free polynomials. We do not know whether this is a new observation. A closely related result of an analytic flavor is [2, Theorem 5.1].
Theorem 8. 17. Let x, y be non-commuting indeterminates and let For any positive intger d, every free polynomial of total degree d in x, y can be written as a polynomial in the d elements u, v 2 , vuv, . . . , vu d−2 v.
Proof. It suffices to prove the result for homogeneous free polynomials. For d ≥ 1 let P d be the complex vector space of homogeneous polynomials of total degree d in x, y and let Sym d be its subspace of symmetric polynomials. Clearly P d has dimension 2 d , and therefore Sym d has dimension 2 d−1 . Let Q d be the space of polynomials in u, v 2 , vuv, . . . , vu d−2 v that are homogeneous of degree d in u, v, and hence also in x, y. Then Q 1 = Cu and Q d ⊆ Sym d . We claim that dim Q d = 2 d−1 = dim Sym d , from which it will follow that Q d = Sym d , as required.
The claim is true when d = 1 since dim Q 1 = 1. Let d > 1 and suppose the claim holds for d − 1. We have By the inductive hypothesis, Hence Q d = Sym d and the theorem follows.
For each γ ∈ Γ and each τ ∈ {−1, 1} γ we define a mapping Φ γτ : U γ → M 3 by the formula where S γτ is the free holomorphic square root on D γ defined in equation (8.11). Let

A Zariski-free atlas for G
The set G defined in the previous subsection is just that, a set. It carries neither a topology nor an atlas of charts that would endow it with a manifold structure. In this section we shall topologize G and equip it with a Zariski-free atlas.
Proposition 8.29 will imply that B is a base for a topology on G. The proof hinges on the following technical statement.
Remark 8.28. The foregoing technical lemma explains the introduction of the Zariski-free topology. It is only because of the exclusion of the varieties V γτ , where a non-generic commutation relation holds, in the definition of the sets U γ that the collection B of Definition 8.21 constitutes a base for a topology on G.
Proposition 8.29. If Ω 1 , Ω 2 ∈ B and ω ∈ Ω 1 ∩ Ω 2 , then there exists we may apply Lemma 8.22 to deduce that Let As (u, x) ∈ U 1 ∩ U 2 , equation (8.30) implies that (u, x, y) ∈ Ω 3 . Clearly To see that Lemma 8.29 implies that B is a base for a unique topology T on G. The topology T is defined so that the maps are homeomorphisms when G γτ carries the T topology and U γ carries the Zariski-free topology. As U γ is a Zariski-free domain, it follows that is a collection of topological nc-co-ordinate patches on G (see condition (4.6)).
In fact A is a Zariski-free atlas for (G, T ). Since Definition (8.20) implies that the sets G γτ cover G, equation (4.4) holds. To see that the transition functions are Zariski-freely holomorphic, observe that We summarize the above observations. Our original hope was to construct a free nc-manifold G and a surjective free holomorphic map π : M 2 → G with the property that, for every symmetric free holomorphic function ϕ on M 2 , there exists a free holomorphic function Φ on G such that ϕ = Φ • π. Our construction above falls short of this goal on at least two counts. Firstly, G is not a free manifold, but a Zariski-free manifold. Secondly, if we make the intended definition then a necessary condition that π(w) ∈ G is that v 2 ∈ I, since we require (u, v 2 ) to belong to some U γ ; thus π does not map M 2 to G. Nonetheless, there is a correspondence between ϕ and a suitable holomorphic function Φ on G, but the correspondence is more subtle, as the next statement shows.
Theorem 8.35. There is a canonical bijection between the classes of (i) symmetric nc functions f that are freely holomorphic on M 2 , and (ii) holomorphic functions F oo defined on the Zariski-free manifold G that are conditionally nc and have the property that, for every w ∈ M 2 , there is a free neighborhood U of w such that F oo is bounded on π(U ) ∩ G. Theorem 8.35 will be proved in the next section, in greater generalitysee Theorem 10.11.

Symmetric free holomorphic functions
In Section 8 we constructed an nc-manifold G for the representation of freely entire symmetric functions on M 2 . In this section we shall do likewise for symmetric free holomorphic functions on an arbitrary symmetric free domain.
Recall that a set S ⊆ M 2 is symmetric if In the sequel S is a fixed symmetric free domain in M 2 . We shall construct a Zariski-free manifold G oo (S) which is, roughly speaking, the restriction of G to S. All the notations and constructions in Section 8 are in effect in this section as well. For w ∈ M 2 we shall frequently employ the change of variables or, equivalently, The operation of tranposition of components will again be denoted by (w 1 , w 2 ) f = (w 2 , w 1 ).
To construct G(S) we need to define several sets and mappings, as indicated schematically in the following figure.
Scheme for the construction of G(S) 9.1 Local inverses of (u, v 2 ) The purpose of this and the next two subsections is to define a submanifold of the Zariski-free manifold G corresponding to a symmetric free domain S in M 2 . First we introduce maps ω γτ which are local inverses of the map w → (u, v 2 ), in the notation of equations (9.1). Recall that, for each γ ∈ Γ, the set W γ defined in equation (8.12) is freely open in M 2 . For each γ ∈ Γ and each τ ∈ {−1, 1} γ we define a mapping ω γτ : W γ → M 2 by the formula where S γτ is the free holomorphic function on D γ defined by equation (8.11).
Solution of these equations for u and x gives u = 1 2 (w 1 + w 2 ) and Thus ω γτ is injective and for all w ∈ ran ω γτ .
Proof. We first show that ran ω γτ is open in the free topology. Equation Once it is known that both the domain and range of ω γτ are freely open sets, that ω γτ is a free biholomorphic mapping follows immediately from the formulas (9.2) and (9.3).
Proof. By Proposition 9.4, ω γτ : W γ → ran ω γτ is a free biholomorphic mapping. It follows by Proposition 7.14 that ω γτ is a homeomorphism when W γ and ran ω γτ are equipped with the Zariski-free topology. Hence equation (9.5) implies that S γτ is Zariski-freely open and equation (9.6) implies that U γτ (S) is Zariski-freely open.
That ω γτ acting on U γτ (S) and ω −1 γτ acting on S γτ are Zariski-free holomorphic mappings (that is, are Zariski-freely locally bounded) follow from the facts that ω γτ acting on W γ and ω −1 γτ acting on ran ω γτ are free holomorphic mappings (and hence are freely locally bounded).

The sets G γτ (S) and G(S)
Recall that for each γ ∈ Γ and each τ ∈ {−1, 1} γ , Φ γτ and G γτ are defined by equations (8.18) and (8.19). For each γ and τ , we define G γτ (S) = Φ γτ (U γτ (S)), and then, following the definition (8.20), we set Recall that the topology T on G was chosen so that the maps Φ γτ : U γ → G γτ are homeomorphisms when U γ carries the Zariski-free topology. Therefore G oo (S) is an open subset of G and as such carries the structure of a Zariskifree manifold. A Zariski-free atlas for G oo (S), A(S), can be obtained from the Zariski-free atlas A for G defined in equation (8.32) by simple restriction of the charts in A to the sets G γτ (S), that is, by the definition We summarize these observations in the following theorem.
We shall use formula (9.14) to extend π to a polynomial map M 2 → M 3 .

9.5
The representation of symmetric functions on S oo Theorem 9.15. Let S be a free symmetric domain in M 2 , G oo (S) the Zariskifree manifold described in Theorem 9.10 and π : S oo → G oo (S) the covering map described in Theorem 9.13. If F oo is a holomorphic function on G oo (S), then f oo = F oo • π is a symmetric Zariski-free holomorphic function on S oo . Conversely, if f oo is a symmetric Zariski-free holomorphic function on S oo , then there exists a unique holomorphic function F oo on G oo (S) such that f oo = F oo • π.
Proof. First assume that F oo is Zariski-freely holomorphic on G oo (S) and let f oo = F oo • π. As F oo and π are Zariski-freely holomorphic, so is f oo = F oo • π.
As π(w f ) = π(w) for all w ∈ S oo , so also f oo (w f ) = f oo (w) for all w ∈ S oo . Thus, f oo is a symmetric Zariski-freely holomorphic function.
Now assume that f oo is a symmetric Zariski-freely holomorphic function on S oo . Attempt to define a function F oo on G oo (S) by the formula F oo (π(w)) = f oo (w), w ∈ S oo .
If π(w 1 ) = π(w 2 ) and w 1 = w 2 then as w 2 = (w f ) 1 and f oo is assumed symmetric, F oo (π(w 1 )) = F oo (π(w 2 )). This proves that F oo is well defined, and clearly f oo = F oo • π. Also, as π is surjective, F oo is defined on all of G oo (S). Since locally on the sets G γτ (S), F oo = f oo • π −1 , a composition of Zariski-freely holomorphic functions, F oo is Zariski-freely holomorphic.

The representation of symmetric functions on S
For any symmetric free domain S in M 2 we define S o ⊆ S by Proof. First assume that w = ω γτ (u, x) ∈ S o , so that 1 2 (w 1 − w 2 ) = S γτ (x) and x ∈ D γ is nonsingular. The spectrum of S γτ (x) is obtained by the application of s γτ to σ(x), which process does not produce two eigenvalues λ and −λ. Hence S γτ (x) ∈ Q, and therefore w ∈ Q. Conversely, suppose that w ∈ S and 1 2 (w 1 − w 2 ) ∈ Q. If we set x = 1 4 (w 1 − w 2 ) 2 and γ = σ(x), then there exists τ ∈ {−1, 1} such that S γτ (x) = 1 2 (w 1 − w 2 ). If u = 1 2 (w 1 + w 2 ), then w = S γτ (u, x) ∈ S o . Let S be a symmetric free domain in M 2 , and define S o by equation (9.16) and S oo by equation (9.12), where π is the polynomial map (9.14). Write G oo for G oo (S), and define G o = π(S o ) and G = π(S). Consider the following diagram: In Theorem 9.15 we showed that f oo : S oo → M 1 is a Zariski-free holomorphic map if and only if there exists a Zariski-free holomorphic map F oo : G oo → M 1 such that F oo • π = f oo . Suppose F : G → M 1 is a free holomorphic map, by which we mean that at every point λ of G there is a free neighborhood B δ of λ in M 3 and a bounded free holomorphic function g on B δ such that g agrees with F on B δ ∩ G. Then F • π is a symmetric free holomorphic function on S.
Question 9.19. Is the converse true?
In this generality, the answer to Question 9.19 is no, as the following example, suggested to us by James Pascoe, shows.
Example 9.20. Let S be the nc-bidisc, that is, the set of pairs of strict contractions in M 2 : The natural choice for an F such that f = F • π is F (z) = 16z 3 (z 2 ) −1 z 3 , but this function is not freely holomorphic on a neighborhood of 0. Indeed, there is no free holomorphic function F defined on a neighborhood B δ of 0 in M 3 such that F • π = f . For suppose that F is such a function. Replacing δ(z) by t(δ(z) − δ(0)), where t ≥ 1/(1 − δ(0) ), we can assume that δ(0) = 0. By [1], F (z) can be represented by a convergent series on B δ whose terms are non-commutative polynomials in the entries δ ij (z). If Expand F in a power series in D, and group terms by homogeneity. Then the left-hand side of equation (9.21) is a linear combination of No linear combination of these three elements is equal to vu 2 v.
One can show more: there is no function F at all satisfying F • π = f on S. We shall show this by giving two points w and W in S ∩ M 2 4 that are identified by π but not by f . Let Let u = U have the (2, 3) and (4, 1) entries equal to 0, but the (2, 3) element of their square not equal to 0. Then π(w) = (u, v 2 , vuv) = π(W ), but since the (1, 4) entry in vu 2 v is r 2 times the (2, 3) entry of u 2 , and in V U 2 V it is the negative of this quantity.
The fact that v is not invertible is crucial in the example. (1) f can be approximated locally uniformly on S in the free topology by a sequence of symmetric free polynomials; (2) Suppose that (i) w 1 , w 2 ∈ S (ii) π(w 1 ) = s −1 π(w 2 )s for some s ∈ I (iii) Proof. (i) Consider any point w 0 ∈ S. Since f is freely holomorphic on S, by Theorem 3.6 there is a basic free neighborhood B δ of w 0 in S and a sequence of free polynomials p n that approximates f uniformly on B δ . Replacing δ(w) by δ(w) ⊕ δ(w f ) if necessary, we can assume that B δ is symmetric, and on replacing p n (w) by 1 2 (p n (w) + p n (w f )), we can assume that p n is symmetric on B δ . (ii) Let Every p n , being symmetric, can be written as a finite linear combination of terms 1, u, and vu j v for j = 0, 1, 2, . . . . By hypothesis, π(w 1 ) = s −1 π(w 2 )s, which means that u 1 = s −1 u 2 s and v 1 u j and so by induction we deduce that p n (w 1 ) = s −1 p n (w 2 )s. Since p n (w 1 ) converges to f (w 1 ) and p n (w 2 ) converges to f (w 2 ), it follows that f (w 1 ) = s −1 f (w 2 )s.
Lemma 9.23. The set of points w ∈ M 2 such that π −1 (π(w)) has cardinality 2 is dense in the finitely open topology.
Proof. Let 2u = w 1 + w 2 and 2v = w 1 − w 2 . The set of w ∈ M 2 n for which v has n distinct non-zero eigenvalues is dense, and the subset where furthermore u has no non-zero entries when expressed with respect to the basis of eigenvectors of v is still dense.
For such a w, if π(w 1 ) = π(w), then u = u 1 , (v) 2 = (v 1 ) 2 and vuv = v 1 uv 1 . The second equation says v 1 has the same eigenvectors as v, with each eigenvalue ± the corresponding eigenvalue for v. If the i th is positive and the j th is negative, then the (i, j) entry of vuv is minus the (i, j) entry of v 1 uv 1 , and is non-zero since u ij is non-zero.
Thus v 1 = ±v, and hence w 1 is either w or w f .

Conditionally nc functions
We shall call a set D ⊂ M d hereditary if, whenever x ⊕ y ∈ D, then x and y are in D. Proof. Let D be a freely open set. Consider z, w such that z ⊕ w ∈ D. Then there is a matrix δ of free polynomials such that z ⊕ w ∈ B δ ⊆ D. Since δ(z ⊕ w) = δ(z) ⊕ δ(w), it follows that z, w ∈ B δ , and hence z, w ∈ D.
Zariski-freely open sets, however, need not be hereditary. such that, for all x ∈ D such that x ⊕ y ∈ D, .
for some a, b, c ∈ C. The unique graded functionf onD such that condition (ii) holds isf (2 ⊕ 1) = b ⊕ c.
The following properties of conditionally nc functions are straightforward to verify. (1) There is a unique graded functionf onD such that equation (9.27) holds.
(2) If f (x) ≤ M for all x ∈ D then f (y) ≤ M for all y ∈D.
(3)f agrees with f on D ∩D.
Remark 9.30. A bounded conditionally nc function f on a free domain U is freely holomorphic. For any x ∈ U we may choose a basic neighborhood B δ of x contained in U ; then f is conditionally nc on B δ , hence is nc on B δ . Thus f is freely locally nc.
On the other hand, we do not know if a freely holomorphic function on a free domain U must be conditionally nc, since being conditionally nc is not a local property. Such a function does preserve direct sums and similarities on basic free neighborhoods, but we do not assert that it preserves similarities on all of U .   [24] in 1770 and independently to Joseph Lagrange [18] in 1798. Lagrange's account is later, but is somewhat more explicit, so we shall call the statement the Waring-Lagrange theorem. It is likely that Euler had the result around the same time.
We now come to our nc version of the theorem, which asserts (to oversimplify somewhat) that symmetric nc functions in 2 variables can be factored through the map π given by the definition (8.34). The precise statement is as follows.
In the theorem we do not assume that S is an nc set. if π(w 1 ) is similar to π(w 2 ) then w 1 ⊕ w 2 ∈ S. There are canonical bijections between the following five sets of graded functions.
(i) Bounded symmetric conditionally nc functions f defined on S.
(ii) Bounded symmetric conditionally nc functions f o defined on S o .
(iii) Bounded symmetric Zariski-freely holomorphic functions f oo defined on S oo that are conditionally nc.
(iv) Bounded holomorphic functions F oo defined on the Zariski-free manifold G oo that are conditionally nc (when G oo is viewed as a subset of M 3 ).
(v) Bounded functions F o defined on G o that are conditionally nc.
Moreover, if functions f, f o , f oo , F o and F oo correspond under these canonical bijections, then the following diagram commutes when the two copies of M 1 are identified.
Proof. Starting with f , one can define f o and f oo by restriction. By Proposition 9.29, f o and f oo are conditionally nc. By Remark 9.30, f is freely holomorphic, and so f oo is Zariski-freely holomorphic, being the restriction of the freely holomorphic function f to a Zariski-freely open set.
(f o → f ) First note that at each level n, the set S \ S o is thin, in the sense of Subsection 7.1. Indeed, to be in S \ S o , by Proposition 9.17, we must have Since f o is bounded and S \ S o is thin, the function f o has an extension by continuity at each level n to a bounded function f on S. By continuity, the function f is symmetric and preserves direct sums, in the sense that if w⊕y ∈ S then w, y ∈ S (since S is a free domain) and f (w⊕y) = f (w)⊕f (y).
Suppose w = s −1 ys for some w, y ∈ S and s ∈ I. Let w n ∈ S o converge to w, and let y n := sw n s −1 . For n large enough, y n ∈ S, and since w n is in S o , so is y n by Proposition 9.17. Then f o (w n ) = s −1 f o (y n )s, and so in the limit we obtain f (w) = s −1 f (y)s.
(f oo → f o ) Starting with f oo , construct f o as follows.
Choose λ ∈ ω γτ (U γ ). Then by Proposition 8.15, λ ⊕ w ∈ ω γτ (U γ ) ⊆ S oo . Thus λ, λ ⊕ w ∈ S oo , which is to say that w ∈ S oo . Thus We claim that The inclusion S o ⊇ S oo is immediate. Consider any y ∈ S o . Then there exists w ∈ S o such that w ⊕ y ∈ S o . By the above construction there exists γ ∈ Γ and τ ∈ {−1, 1} γ such that w ⊕ y ∈ ω γτ (W γ ). Let λ ∈ ω γτ (U γ ); then, again by Proposition 8.15, λ ⊕ w and λ ⊕ w ⊕ y belong to S oo . Hence y ∈ S oo . Define f o to be the restriction of f oo to S o . By Proposition 9.29(3), f o is an extension of f oo .
Since f oo is conditionally nc, there is a function f oo on S oo such that As f oo is bounded, so is f o , by Proposition 9.29 (2) and, since f oo is symmetric, so is f o .
To show that f o is conditionally nc we must exhibit a function f o on S o such that, for all y ∈ S o and all w ∈ S o such that w ⊕ y ∈ S o , In view of equation (10.4), we may define f o to be f oo . Then we must show that whenever w, w ⊕ y ∈ S o . As above there exists λ ∈ S oo such that λ ⊕ w ⊕ y ∈ S oo , which means that λ ⊕ w is also in S oo . Thus It follows that equation (10.7) holds, as required.
To see that f o preserves similarities, we argue as follows. Let w, s −1 ws = y ∈ S o . By assumption (10.2) we have w ⊕ y ∈ S. As w and y have the same spectrum, By Proposition 9.17, w ⊕ y ∈ S o . Hence there exists λ ∈ S oo such that λ ⊕ w ⊕ y is in S oo , as are λ ⊕ w and λ ⊕ y. Since f oo preserves similarities and We claim that if π(w 1 ) = π(w 2 ) for some pair w 1 , w 2 in S o , then f (w 1 ) = f (w 2 ). Indeed, by assumption (10.2), w 1 ⊕ w 2 ∈ S. By the symmetry of S, (w 1 ⊕ w 2 ) f ∈ S, which is to say that w f 1 ⊕ w f 2 ∈ S. By the symmetry of π, π(w 1 ⊕ w 2 ) = π((w 1 ⊕ w 2 ) f ). Again by assumption (10.2), w 1 ⊕ w 2 ⊕ w f 1 ⊕ w f 2 ∈ S. Furthermore, since w 1 ∈ S o , it follows from Proposition 9.17 that w 1 1 − w 2 1 is invertible. We may therefore apply Lemma 9.22 to conclude that f (w 1 ) = f (w 2 ). Hence f o (w 1 ) = f o (w 2 ).
Thus, at the level of set theory, we can define F o by f o = F o • π. To see that F o is conditionally nc, suppose first that π(w) and π(w) ⊕ π(y) are in G o , for some w, w ⊕ y ∈ S o . Then y ∈ S o also, by Proposition 9.17. Thus To show that F o preserves similarities, consider similar triples z 1 and z 2 = sz 1 s −1 in G o . Then there exist w 1 , w 2 ∈ S o such that z 1 = π(w 1 ), z 2 = π(w 2 ). We have Not knowing that sw 1 s −1 ∈ S o , we use Lemma 9.22. By Assumption (10.2), w 1 ⊕ w 2 ⊕ w f 1 ⊕ w f 2 ∈ S, and since w 1 ∈ S o , the lemma tells us that f (w 2 ) = s −1 f (w 1 )s, and so F o (z 2 ) = s −1 F o (z 1 )s, as required. ( Then f o is bounded, symmetric, and conditionally nc. The set S o is not closed with respect to direct sums, that is, it is not an nc set. If S in Theorem 10.1 is assumed to be a symmetric freely open nc set, then assumption (10.2) is automatically satisfied, and the conditionally nc functions on S are the same as the nc functions. This yields the following corollary. If we drop the assumption that f be bounded, and require only that f be freely locally bounded, then this imposes corresponding restrictions on the other functions. Recall that in Definition 7.1 we defined a function ϕ defined on U \ T to be locally bounded on U if, for every point z in U , there is a neighborhood V of z such that ϕ is bounded on V \T . The theorem becomes: There are canonical bijections between the following five sets of graded functions.
(i) Symmetric conditionally nc functions f that are freely holomorphic on S.
(ii) Symmetric conditionally nc functions f o defined on S o that are freely locally bounded on S.
(iii) Symmetric Zariski-freely holomorphic functions f oo defined on S oo that are conditionally nc and freely locally bounded on S.
(iv) Holomorphic functions F oo defined on the Zariski-free manifold G oo that are conditionally nc and have the property that for all w in S, there is a free neighborhood U of w such that F oo is bounded on π(U ) ∩ G oo .
(v) Bounded graded functions F o defined on G o that are conditionally nc and have the property that for all w in S, there is a free neighborhood U of w such that F o is bounded on π(U ) ∩ G o .
Moreover, if functions f, f o , f oo , F o and F oo correspond under these canonical bijections, then the diagram (10.3) commutes when the two copies of M 1 are identified.

Nc Newton-Girard formulae
Instances of the Waring-Lagrange theorem are furnished by a series of formulae for power sums in terms of elementary symmetric functions. Classically such formulae were first given in 1629 by Albert Girard [11], though they were subsequently often attributed to Newton [19]. We are only concerned with polynomials in two variables x and y. When these variables commute, the Newton-Girard formulae express the power sums p n (x, y) def = x n + y n in terms of the elementary symmetric functions e 1 (x, y) = x + y, e 2 (x, y) = xy.
Further formulae are obtained from the recursion p n+2 = e 1 p n+1 − e 2 p n .
In the case of non-commuting indeterminates x, y we retain the notation p n for the nth power sum, but now there is no finite set of 'elementary symmetric functions' in terms of which all p n can be written as polynomials. However, the foregoing nc Waring-Lagrange theorems tell us that it is possible to express p n as a rational expression in the variables α = u, β = v 2 , γ = vuv (11.1) where u = 1 2 (x + y), v = 1 2 (x − y). In this section we shall show how to construct such expressions explicitly, in the spirit of Girard and Newton. Since we are obliged to work with rational expressions, it is natural to allow the n in p n to be an arbitrary integer, positive, negative or zero. We first express p n and the antisymmetric rational function q n def = x n − y n in terms of u and v. We have p 0 = 2, q 0 = 0. For any integer n, p n = xx n−1 + yy n−1 = (u + v)x n−1 + (u − v)y n−1 = u(x n−1 + y n−1 ) + v(x n−1 − y n−1 ) = up n−1 + vq n−1 . We may write equations (11.2) and (11.3) in the matrix form p n q n = T p n−1 q n−1 for all n ∈ Z (11.4) where T = u v v u . (11.5) Define the free polynomial s n even in u, v for n ≥ 0 to be the sum of all monomials in u, v of total degree n and of even degree in v. When regarded as a polynomial in x, y, s n even is symmetric. Likewise, s n odd is defined to be the sum of all monomials in u, v of total degree n and odd degree in v. Thus s n odd is antisymmetric as a polynomial in x, y.
In general, any monomial in which v occurs with even degree can be written as a monomial in α, β, γ and β −1 . Indeed, starting at one end of the monomial, replace all the initial u's by α's. The first v must be followed by another (since the number of v's is even). If it is immediately following, replace v 2 by β. If there are k u's between the first and second v's, replace vu k v by (γβ −1 ) k−1 γ. Continue in this way until all u's and v's have been replaced.
We have shown the following.
Thus p −n = 2f n (u, v) in the notation of Lemma 11.16. By that lemma, p −n is expressible as a rational function of the variables α, β, γ.
One can readily calculate the first two P −n from these formulae; we expect that P −2 can be further simplified. P −1 (α, β, γ) = 2(α − βγ −1 β) −1 (11.20) P −2 (α, β, γ) = 2 α 2 + β − αβ(β −1 γ + γ −1 β 2 ) −1 − γ(γ −1 βγ + β 2 ) −1 γ − αβ(γβ −1 γ + β 2 ) −1 βα −(β −1 γβ −1 + βγ −1 ) −1 α −1 . (11.21) There are some minor subtleties concerning the interpretation of the foregoing Newton-Girard formulae for p n . When n ≥ 0, since P n is a free polynomial in α, β, γ, β −1 , the statement x n + y n = P n (α, β, γ) is meaningful and valid whenever x and y are matrices of the same order such that x − y is nonsingular. It can also be interpreted as an identity in the free field, which is the smallest universal division ring containing the ring of free polynomials in x, y. When n < 0 the issue is less immediate, since then the structure of P n is more complicated. In this paper we are concerned with functions of tuples of matrices. In this context, a noncommutative rational expression is called non-degenerate if its domain in M d is non-empty. The domain will always be Zariski open at every level n (restrictions on the domain come about when there is an inverse in the expression, as whatever needs to be inverted must be non-singular). Different nondegenerate noncommutative rational expressions may have different domains where they can be evaluated, but agree on the intersection of these domains. Such expressions are called equivalent, and a noncommutative rational function is formally an equivalence class of nondegenerate noncommutative rational expressions. See for example [23] for a discussion. For the expressions P −n it is easy to see that they are non-degenerate; from equation (11.19) we see that the functions can be evaluated as long as all four of the expressions α − βγ −1 β, β − γβ −1 α, β − αβ −1 γ, γ − βα −1 β are invertible. In particular, choice of x, y as the scalar matrices 4 and 2 respectively gives the values α = 3, β = 1, γ = 3, and one finds that the above four expressions do evaluate to invertible matrices.
It is interesting to compare the identities in Theorems 11.10 and 11.18, thought of as equations in the algebra of rational functions in x and y, with the statements about nc functions contained in our main theorems in Sections 8 and 9. When n ≥ 0 the symmetric free polynomial p n is freely holomorphic on M 2 . Theorem 8.35 applies to yield a holomorphic function P n on the Zariski-free manifold G, having a certain local boundedness property and satisfying p n = P n • π on a suitable subset of M 2 . When n < 0 we must take the domain S of p n to be the set {(x, y) ∈ M 2 : x ∈ I, y ∈ I}. Theorem 8.35 no longer applies, so we appeal to Theorem 10.1. Notice that here S is an nc set, so that Assumption (10.2) is automatically satisfied. We again deduce that there is a holomorphic function P n , this time on the Zariski-free manifold G oo (S), satisfying a version of the relation p n = P n • π.