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

Conservation laws and symmetries of difference equations

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Elimination Theory for differential difference polynomials

In this paper we give an elimination algorithm for differential difference polynomial systems. We use the framework of a generalization of Ore algebras, where the independent variables are non-commutative. We prove that for certain term orderings, Buchberger's algorithm applied to differential difference systems terminates and produces a Gröbner basis. Therefore, differential-difference algebras provide a new instance of non-commutative graded rings which are effective Gröbner structures