Bounds and definability in polynomial rings

Bounds and Definability in Polynomial Rings

We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor. In particular, we show that the module of syzygies of polynomials f1, . . . , fn ∈ R[X1, . . . , XN ] with coefficients in a Prüfer domain R can be generated by elements whose degrees are bounded by a number only depending on N ...

Bounds in Polynomial Rings over Artinian Local Rings

Let R be a (mixed characteristic) Artinian local ring of length l and let X be an n-tuple of variables. This paper provides bounds over the ring R[X] on the degrees of the output of several algebraic constructions in terms of l, n and the degrees of the input. For instance, if I is an ideal in R[X] generated by polynomials gi of degree at most d and if f is a polynomial of degree at most d belo...

Polynomial Bounds for Rings of Invariants

Hilbert proved that invariant rings are finitely generated for linearly reductive groups acting rationally on a finite dimensional vector space. Popov gave an explicit upper bound for the smallest integer d such that the invariants of degree ≤ d generate the invariant ring. This bound has factorial growth. In this paper we will give a bound which depends only polynomially on the input data.

Nonstandard Methods for Bounds in Differential Polynomial Rings

Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt [“Bounds in the theory of polynomial rings over fields. A nonstandard approach.”, Inventionnes Mathematicae, 76:77–91, 1984] are here extended to differential polynomial rings over differential fields. Amon...

Differential Polynomial Rings of Triangular Matrix Rings