Signature Gr\"obner bases in free algebras over rings

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal domain. We give an algorithm for computing them, combining elements from the theory of commutative and noncommutative (signature) Gr\"obner bases, and prove its correctness. Applications include extensions of the free algebra with commutative variables, e.g., for homogenization purposes or for performing ideal theoretic operations such as intersections, and computations over $\mathbb{Z}$ as universal proofs over fields of arbitrary characteristic. By extending the signature cover criterion to our setting, our algorithm also lifts some technical restrictions from previous noncommutative signature-based algorithms, now allowing, e.g., elimination orderings. We provide a prototype implementation for the case when $R$ is a field, and show that our algorithm for the mixed algebra is more efficient than classical approaches using existing algorithms.Comment: 10 page

A Gröbner-Basis Theory for Divide-and-Conquer Recurrences

International audienceWe introduce a variety of noncommutative polynomials that represent divide-and-conquer recurrence systems. Our setting involves at the same time variables that behave like words in purely noncom-mutative algebras and variables governed by commutation rules like in skew polynomial rings. We then develop a Gröbner-basis theory for left ideals of such polynomials. Strikingly, the nature of commutations generally prevents the leading monomial of a polynomial product to be the product of the leading monomials. To overcome the difficulty, we consider a specific monomial ordering, together with a restriction to monic divisors in intermediate steps. After obtaining an analogue of Buchberger's algorithm, we develop a variant of the 4 algorithm, whose speed we compare

Letterplace ideals and non-commutative Gröbner bases

In this paper we propose a 1-to-1 correspondence between graded two-sided ideals of the free associative algebra and some class of ideals of the algebra of polynomials, whose variables are double-indexed commuting ones. We call these ideals the "letterplace analogues" of graded two-sided ideals. We study the behaviour of the generating sets of the ideals under this correspondence, and in particular that of the Grobner bases. In this way, we obtain a new method for computing non-commutative homogeneous Grobner bases via polynomials in commuting variables. Since the letterplace ideals are stable under the action of a monoid of endomorphisms of the polynomial algebra, the proposed algorithm results in an example of a Buchberger procedure "reduced by symmetry". Owing to the portability of our algorithm to any computer algebra system able to compute commutative Grobner bases, we present an experimental implementation of our method in SINGULAR. By means of a representative set of examples, we show finally that our implementation is competitive with computer algebra systems that provide non-commutative Grobner bases from classical algorithms