Noncommutative Polynomials Research Articles

Boolean cumulants and subordination in free probability

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation [Formula: see text] for free random variables [Formula: see text] and a Borel function [Formula: see text] is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form [Formula: see text]. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

A note on the free and cyclic differential calculus

In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu's fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. On the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples, such as a discrete version of the It\^o stochastic integral, can be treated.

Efficient Evaluation of Noncommutative Polynomials Using Tensor and Noncommutative Waring Decompositions

This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial p with respect to the goal of evaluating p efficiently on tuples of matrices. Such a decomposition can reduce the number of matrix multiplications needed to evaluate a noncommutative polynomial and is valuable when a single polynomial must be evaluated on many matrix tuples. In pursuit of this goal we examine a noncommutative analog of the classical Waring problem and various related decompositions. For example, we consider a “Waring decomposition” in which each product of linear terms is actually a power of a single linear NC polynomial or more generally a power of a homogeneous NC polynomial. We describe how NC polynomials compare to commutative ones with regard to these decompositions, describe a method for computing the NC decompositions and compare the effect of various decompositions on the speed of evaluation of generic NC polynomials.

A standard form in (some) free fields: How to construct minimal linear representations

AbstractWe describe a standard form for the elements in the universal field of fractions of free associative algebras (over acommutativefield). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in a standard form). This completes “minimal” arithmetic in free fields since “minimal” constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.

On the rational relationships among pseudo-roots of a non-commutative polynomial

For a non-commutative ring R , we consider factorizations of polynomials in R [ t ] where t is a central variable. A pseudo-root of a polynomial p ( t ) = p 0 + p 1 t + ⋯ p k t k is an element ξ ∈ R , for which there exist polynomials q 1 , q 2 such that p = q 1 ( t − ξ ) q 2 . We investigate the rational relationships that hold among the pseudo-roots of p ( t ) by using the diamond operations for cover graphs of modular lattices . When R is a division ring, each finite subset S of R corresponds to a unique minimal monic polynomial f S that vanishes on S . By results of Leroy and Lam [16] , the set of polynomials { f T : T ⊆ S } with the right-divisibility order forms a lattice with join operation corresponding to (left) least common multiple and meet operation corresponding to (right) greatest common divisor. The set of edges of the cover graph of this lattice correspond naturally to a set Λ S of pseudo-roots of f S . Given an arbitrary subset of Λ S , our results provide a graph theoretic criterion that guarantees that the subset rationally generates all of Λ S , and in particular, rationally generates S .

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.

Multilinear polynomials are surjective on algebras with surjective inner derivations

Let f(X1,…,Xn) be a nonzero multilinear noncommutative polynomial. If A is a unital algebra with a surjective inner derivation, then every element in A can be written as f(a1,…,an) for some ai∈A.

On Finite Noncommutative Gröbner Bases

It is well known that in the noncommutative polynomial ring in serveral variables Buchberger’s algorithm does not always terminate. Thus, it is important to characterize noncommutative ideals that admit a finite Gröbner basis. In this context, Eisenbud, Peeva and Sturmfels defined a map γ from the noncommutative polynomial ring k⟨X1, …, Xn⟩ to the commutative one k[x1, …, xn] and proved that any ideal [Formula: see text] of k⟨X1, …, Xn⟩, written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn], amits a finite Gröbner basis with respect to a special monomial ordering on k⟨X1, …, Xn⟩. In this work, we approach the opposite problem. We prove that under some conditions, any ideal [Formula: see text] of k⟨X1, …, Xn⟩ admitting a finite Gröbner basis can be written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn].

Commutators and images of noncommutative polynomials

Let A be an algebra and let f be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between [A,A], the linear span of commutators in A, and spanf(A), the linear span of the image of f in A. In particular, we show that [A,A]=A implies spanf(A)=A. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if C is a commutative unital algebra over a field F of characteristic 0, A is the matrix algebra Mn(C), and the polynomial f is neither an identity nor a central polynomial of Mn(F), then every commutator in A can be written as a difference of two elements, each of which is a sum of 7788 elements from f(A) (if C=F is an algebraically closed field, then 4 elements suffice). Similar results are obtained for some other algebras, in particular for the algebra B(H) of all bounded linear operators on a Hilbert space H.

A Noncommutative Cycle Index and New Bases of Quasi-symmetric Functions and Noncommutative Symmetric Functions

We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to QSym. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of \(\mathrm{FQSym}\) induced by the pattern-replacement relation \(321\equiv 231\) and \(312\equiv 132\).

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set ${\rm sl}_n(K)$ of matrices of trace 0, or all of $M_n(K)$. This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for $n=2$ in Section 2, some decisive results for $n=3$ in Section 3, and partial information for $n\geq 3$ in Section 4, also for non-multilinear polynomials. In addition we consider the case of $K$ not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.

Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information

This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information, highlighting the special case of Lipschitz conjugate variables. For the first time in this generality, it is shown that the analytic distributions of those evaluations have Hölder continuous cumulative distribution functions with an explicit Hölder exponent that depends only on the degree of the considered polynomial. For linear polynomials, we reach in the case of finite non-microstates free Fisher information the optimal Hölder exponent 23, and get Lipschitz continuity in the case of Lipschitz conjugate variables. In particular, our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy, which partially settles a conjecture of Charlesworth and Shlyakhtenko [8].We further provide a very general criterion that gives for weak approximations of measures having Hölder continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance.Finally, we combine these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general Gibbs laws. For Gibbs laws, this extends the corresponding result obtained in [21] from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of [23,24] in terms of the Kolmogorov distance.

A FACTORIZATION THEORY FOR SOME FREE FIELDS

Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level of minimal linear representations . We establish a factorization theory by providing an alternative definition of left (and right) divisibility based on the rank of an element and show that it coincides with the " classical'' left (and right) divisibility for non-commutative polynomials. Additionally we present an approach to factorize elements, in particular rational formal power series, into their (generalized) atoms. The problem is reduced to solving a system of polynomial equations with commuting unknowns.

Noncommutative Polynomials Describing Convex Sets

The free closed semialgebraic set $D_f$ determined by a hermitian noncommutative polynomial $f$ is the closure of the connected component of $\{(X,X^*)\mid f(X,X^*)>0\}$ containing the origin. When $L$ is a hermitian monic linear pencil, the free closed semialgebraic set $D_L$ is the feasible set of the linear matrix inequality $L(X,X^*)\geq 0$ and is known as a free spectrahedron. Evidently these are convex and it is well-known that a free closed semialgebraic set is convex if and only it is a free spectrahedron. The main result of this paper solves the basic problem of determining those $f$ for which $D_f$ is convex. The solution leads to an efficient algorithm that not only determines if $D_f$ is convex, but if so, produces a minimal hermitian monic pencil $L$ such that $D_f=D_L$. Of independent interest is a subalgorithm based on a Nichtsingul\"arstellensatz presented here: given a linear pencil $L'$ and a hermitian monic pencil $L$, it determines if $L'$ takes invertible values on the interior of $D_L$. Finally, it is shown that if $D_f$ is convex for an irreducible hermitian polynomial $f$, then $f$ has degree at most two, and arises as the Schur complement of an $L$ such that $D_f=D_L$.

Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra

Abstract This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots ,x_g)$ is $\mathscr{Z}(\,f)=(\mathscr{Z}_n(\,f))_n$, where $\mathscr{Z}_n(\,f)=\{X \in{\operatorname{M}}_{n}({\mathbb{C}})^g \colon \det f(X) = 0\}.$ The 1st main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $\mathscr{Z}_n(\,f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $\mathscr{Z}^{{\operatorname{re}}}(h)=\{X\colon \det h(X,X^*)=0\}$. A polynomial $f$ that depends on $x$ alone (no $x^*$ variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic $f$ but for $h$ dependent on possibly both $x$ and $x^*$, the containment $\mathscr{Z}(\,f) \subseteq \mathscr{Z}^{{\operatorname{re}}} (h)$ is equivalent to each factor of $f$ being “stably associated” to a factor of $h$ or of $h^*$. For perspective, classical Hilbert-type Nullstellensätze typically apply only to analytic polynomials $f,h $, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above “algebraic certificate” does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018): 589–626) obtained such a theorem for special classes of analytic polynomials $f$ and $h$. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros ${\mathcal{V}}(\,f)=\{X\colon f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.

Local theory of free noncommutative functions: germs, meromorphic functions, and Hermite interpolation

Free analysis is a quantization of the usual function theory much like operator space theory is a quantization of classical functional analysis. Basic objects of free analysis are noncommutative functions. These are maps on tuples of matrices of all sizes that preserve direct sums and similarities. This paper investigates the local theory of noncommutative functions. The first main result shows that for a scalar point Y Y , the ring O Y ua \mathcal {O}^{\operatorname {ua}}_Y of uniformly analytic noncommutative germs about Y Y is an integral domain and admits a universal skew field of fractions, whose elements are called meromorphic germs. A corollary is a local-global rank principle that connects ranks of matrix evaluations of a matrix A A over O Y ua \mathcal {O}^{\operatorname {ua}}_Y with the factorization of A A over O Y ua \mathcal {O}^{\operatorname {ua}}_Y . Different phenomena occur for a semisimple tuple of nonscalar matrices Y Y . Here it is shown that O Y ua \mathcal {O}^{\operatorname {ua}}_Y contains copies of the matrix algebra generated by Y Y . In particular, there exist nonzero nilpotent uniformly analytic functions defined in a neighborhood of Y Y , and O Y ua \mathcal {O}^{\operatorname {ua}}_Y does not embed into a skew field. Nevertheless, the ring O Y ua \mathcal {O}^{\operatorname {ua}}_Y is described as the completion of a free algebra with respect to the vanishing ideal at Y Y . This is a consequence of the second main result, a free Hermite interpolation theorem: if f f is a noncommutative function, then for any finite set of semisimple points and a natural number L L there exists a noncommutative polynomial that agrees with f f at the chosen points up to differentials of order L L . All the obtained results also have analogs for (nonuniformly) analytic germs and formal germs.

Free Bertini’s theorem and applications

The simplest version of Bertini’s irreducibility theorem states that the generic fiber of a noncomposite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if f f is a noncommutative polynomial such that f − λ f-\lambda factors for infinitely many scalars λ \lambda , then there exist a noncommutative polynomial h h and a nonconstant univariate polynomial p p such that f = p ∘ h f=p\circ h . Two applications of free Bertini’s theorem for matrix evaluations of noncommutative polynomials are given. An eigenlevel set of f f is the set of all matrix tuples X X where f ( X ) f(X) attains some given eigenvalue. It is shown that eigenlevel sets of f f and g g coincide if and only if f a = a g fa=ag for some nonzero noncommutative polynomial a a . The second application pertains to quasiconvexity and describes polynomials f f such that the connected component of \{X \text { tuple of symmetric n×n matrices}\colon \lambda I\succ f(X) \} about the origin is convex for all natural n n and λ > 0 \lambda >0 . It is shown that such a polynomial is either everywhere negative semidefinite or the composition of a univariate and a convex quadratic polynomial.

The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

We develop a non-commutative polynomial version of the invariant theory of the quantum general linear supergroup Uq(glm|n). A non-commutative Uq(glm|n)-module superalgebra Pr|sk|l is constructed, which is the quantum analogue of the supersymmetric algebra over Ck|l⊗Cm|n⊕Cr|s⊗(Cm|n)⁎. We analyse the structure of the subalgebra of Uq(glm|n)-invariants in Pr|sk|l by using a quantum super analogue of Howe duality.The subalgebra of Uq(glm|n)-invariants in Pr|sk|l is shown to be finitely generated. We determine its generators and establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto it. This establishes the first fundamental theorem of invariant theory for Uq(glm|n).We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if m⩾min⁡{k,r} and n⩾min⁡{l,s}, and obtain a PBW basis for the subalgebra of invariants in this case. When the homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel. This way we obtain the relations obeyed by the generators of the subalgebra of invariants, producing the second fundamental theorem of invariant theory for Uq(glm|n).We consider the special case n=0 in greater detail, obtaining a complete treatment of the non-commutative polynomial version of the invariant theory of Uq(glm). In particular, the explicit SFT proved here is believed to be new. We also recover the FFT and SFT of invariant theory for the general linear superalgebra from the classical limit (i.e. q→1) of our results.

The images of noncommutative polynomials evaluated on the quaternion algebra

Let [Formula: see text] be a multilinear polynomial in several noncommuting variables with coefficients in an arbitrary field [Formula: see text]. Kaplansky conjectured that for any [Formula: see text], the image of [Formula: see text] evaluated on the set [Formula: see text] of [Formula: see text] by [Formula: see text] matrices is a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra.

Noncommutative Partial Convexity Via $$\Gamma $$-Convexity

Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a tuple of symmetric polynomials $\Gamma$, a free set is called $\Gamma$-convex if it closed under isometric conjugation by isometries intertwining $\Gamma$. We establish an Effros-Winkler Hahn-Banach separation theorem for $\Gamma$-convex sets; they are delineated by linear pencils in the coordinates of $\Gamma$ and the variables $x$.