Michael Wibmer

Gröbner bases for polynomial systems with parameters

Grobner bases are the computational method par excellence for studying polynomial systems. In the case of parametric polynomial systems one has to determine the reduced Grobner basis in dependence of the values of the parameters. In this article, we present the algorithm GrobnerCover which has as inputs a finite set of parametric polynomials, and outputs a finite partition of the parameter space into locally closed subsets together with polynomial data, from which the reduced Grobner basis for a given parameter point can immediately be determined. The partition of the parameter space is intrinsic and particularly simple if the system is homogeneous.

σ-Galois Theory of Linear Difference Equations

We develop a Galois theory for systems of linear difference equations with an action of an endomorphism {\sigma}. This provides a technique to test whether solutions of such systems satisfy {\sigma}-polynomial equations and, if yes, then characterize those. We also show how to apply our work to study isomonodromic difference equations and difference algebraic properties of meromorphic functions.

Difference algebraic relations among solutions of linear differential equations

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski and Y. Peterzil.

Skolem–Mahler–Lech type theorems and Picard–Vessiot theory

We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem-Mahler-Lech theorem to rational function coefficients. The second problem is the question whether or not for a given linear difference equation there exists a Picard-Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell-Lang conjecture. This allows us to deduce solutions to the first two problems in a particular but fairly general special case.

Tannakian Categories with Semigroup Actions

Ostrowskis theorem implies that

Leakage quantification of cryptographic operations

\log(x),\log(x+1),\ldots

On Invariant Relations between Zeros of Polynomials

are algebraically independent over

Software for discussing parametric polynomial systems : the Gröbner cover

\mathbb{C}(x)

Existence of ∂-parameterized Picard–Vessiot extensions over fields with algebraically closed constants

. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution

Gröbner bases for families of affine or projective schemes

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