Certifying operator identities via noncommutative Gröbner bases

Abstract

Matrices or linear operators and their identities can be modelled algebraically by noncommutative polynomials in the free algebra. For proving new identities of matrices or operators from given ones, computations are done formally with noncommutative polynomials. Computations in the free algebra, however, are not necessarily compatible with formats of matrices resp. with domains and codomains of operators. For ensuring validity of such computations in terms of operators, in principle, one would have to inspect every step of the computation. In [9], an algebraic framework is developed that allows to rigorously justify such computations without restricting the computation to compatible expressions. The main result of that paper reduces the proof of an operator identity to verifying membership of the corresponding polynomial in the ideal generated by the polynomials corresponding to the assumptions and verifying compatibility of this polynomial and of the generators of the ideal.

Computing elements of certain form in ideals to prove properties of operators

“…We note that Theorem 2.2 also applies in the same way to right ideals. For further information on Gröbner bases in the free algebra, see for example [14,20,15,9], and also [11] for an overview on available software and further references.…”

“…All the procedures presented here are implemented in the Mathematica software package OperatorGB [9], which also provides support for proving properties of matrices and operators along these lines using the framework mentioned above. The package is available at https://clemenshofstadler.com/ software/ along with a notebook containing detailed proofs of the examples discussed in this paper.…”

Signature Gröbner bases, bases of syzygies and cofactor reconstruction in the free algebra

“…We also provide a toy implementation 1 of the algorithms presented in this paper in the Mathematica package OperatorGB [HRR19,Hof20]. Additionally, we show experimentally that the use of signatures allows to drastically reduce the number of S-polynomials considered and reduced to zero.…”

“…The problem of computing the cofactors of a Gröbner basis is also central when working with noncommutative polynomials. In particular, when proving operator identities, this information allows to construct a proof certificate for a given identity, which can be checked easily and independently of how it was obtained, see for example [Hof20]. Like in the commutative case, the classical theory and algorithms for computing Gröbner bases in modules can also be used to obtain such information [BK06,Mor16], but those algorithms are significantly more expensive than a mere Gröbner basis computation.…”

“…In the following, we briefly recall the most important results about Gröbner bases in K X . For a more extensive treatment of this subject, we refer to the recent surveys [Xiu12,Mor16,Hof20]. The main concept needed to discuss and compute noncommutative Gröbner bases is polynomial reduction.…”