Improved Polynomial Remainder Sequences for Ore Polynomials

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients

On sequences associated to the invariant theory of rank two simple Lie algebras

We study two families of sequences, listed in the On-Line Encyclopedia of Integer Sequences (OEIS), which are associated to invariant theory of Lie algebras. For the first family, we prove combinatorially that the sequences A059710 and A108307 are related by a binomial transform. Based on this, we present two independent proofs of a recurrence equation for A059710, which was conjectured by Mihailovs. Besides, we also give a direct proof of Mihailovs' conjecture by the method of algebraic residues. As a consequence, closed formulae for the generating function of sequence A059710 are obtained in terms of classical Gaussian hypergeometric functions. Moreover, we show that sequences in the second family are also related by binomial transforms

Algorithms for discrete differential equations of order 1

To appear in ISSAC'22International audienceDiscrete differential equations of order 1 are equations relating polynomially $F(t,u)$, a power series in $t$ with polynomial coefficients in a ``catalytic'' variable~$u$, and one of its specializations, say~$F(t,1)$. Such equations are ubiquitous in combinatorics, notably in the enumeration of maps and walks. When the solution $F$ is unique, a celebrated result by Bousquet-M\'elou, reminiscent of Popescu's theorem in commutative algebra, states that $F$ is \emph{algebraic}. We address algorithmic and complexity questions related to this result. In \emph{generic} situations, we first revisit and analyze known algorithms, based either on polynomial elimination or on the guess-and-prove paradigm. We then design two new algorithms: the first has a geometric flavor, the second blends elimination and guess-and-prove. In the \emph{general} case (no genericity assumptions), we prove that the total arithmetic size of the algebraic equations for $F(t,1)$ is bounded polynomially in the size of the input discrete differential equation, and that one can compute such equations in polynomial time

Heun functions and diagonals of rational functions (unabridged version)

International audienceWe provide a set of diagonals of simple rational functions of four variables that are seen to be squares of Heun functions. Each time, these Heun functions, obtained by creative telescoping, turn out to be pullbacked 2 F 1 hypergeometric functions and in fact classical modular forms. We even obtained Heun functions that are automorphic forms associated with Shimura curves as solutions of telescopers of rational functions