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.
•We illustrate a recent method for proving identities of matrices and linear operators.•Computations are done in an abstract ring ignoring domains and codomains.•The theory ensures that identities ...are also valid in terms of matrices or operators.•We prove and generalize Hartwig’s result on three Moore-Penrose inverses.•Both hand-made and computer-assisted proofs with the package OperatorGB are discussed.
When improving results about generalized inverses, the aim often is to do this in the most general setting possible by eliminating superfluous assumptions and by simplifying some of the conditions in statements. In this paper, we use Hartwig’s well-known triple reverse order law as an example for showing how this can be done using a recent framework for algebraic proofs and the software package OperatorGB. Our improvements of Hartwig’s result are proven in rings with involution and we discuss computer-assisted proofs that show these results in other settings based on the framework and a single computation with noncommutative polynomials.
Short proofs of ideal membership Hofstadler, Clemens; Verron, Thibaut
Journal of symbolic computation,
November-December 2024, 2024-11-00, Letnik:
125
Journal Article
Recenzirano
Odprti dostop
A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not ...unique, and some can be a lot more complicated than others. In this work, we consider the problem of computing sparsest cofactor representations, i.e., representations with a minimal number of terms, of a given element in a polynomial ideal. While we focus on the more general case of noncommutative polynomials, all results also apply to the commutative setting.
We show that the problem of computing cofactor representations with a bounded number of terms is decidable and NP-complete. Moreover, we provide a practical algorithm for computing sparse (not necessarily optimal) representations by translating the problem into a linear optimization problem and by exploiting properties of signature-based Gröbner basis algorithms. We show that, for a certain class of ideals, representations computed by this method are actually optimal, and we present experimental data illustrating that it can lead to noticeably sparser cofactor representations.
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.
Signature-based algorithms have become a standard approach for computing Gröbner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of ...noncommutative polynomials in the free algebra. In this paper, we present a signature-based algorithm for computing Gröbner bases in precisely this setting. The algorithm is an adaptation of Buchberger's algorithm including signatures. We prove that our algorithm correctly enumerates a signature Gröbner basis as well as a Gröbner basis of the module generated by the leading terms of the generators' syzygies, and that it terminates whenever the ideal admits a finite signature Gröbner basis. Additionally, we adapt well-known signature-based criteria eliminating redundant reductions, such as the syzygy criterion, the F5 criterion and the singular criterion, to the case of noncommutative polynomials. We also generalise reconstruction methods from the commutative setting that allow to recover, from partial information about signatures, the coordinates of elements of a Gröbner basis in terms of the input polynomials, as well as a basis of the syzygy module of the generators. We have written a toy implementation of all the algorithms in the Mathematica package OperatorGB and we compare our signature-based algorithm to the classical Buchberger algorithm for noncommutative polynomials.
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