Bounds for algorithms in differential algebra

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal. We also give an algorithm for converting a characteristic decomposition of a radical differential ideal from one ranking into another. This algorithm performs all differentiations in the beginning and then uses a purely algebraic decomposition algorithm.Comment: 40 page

On the formula for the PI-exponent of Lie algebras

We prove that one of the conditions in M.V. Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite dimensional semisimple Lie algebra acts by derivations on a finite dimensional Lie algebra over a field of characteristic $0$, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial $G$-identities of a finite dimensional Lie algebra with a rational action of a connected reductive affine algebraic group $G$ by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for $H$-module Lie algebras whose solvable radical is nilpotent, assuming only the $H$-invariance of the radical, i.e. under weaker assumptions on the $H$-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for $G$-codimensions of all finite dimensional Lie $G$-algebras whose solvable radical is nilpotent, for an arbitrary group $G$.Comment: 15 pages. Section 2 (def. of the free H-alg., H-ident., and H-codim.) is the same as Subs. 1.3 in arXiv:1207.1699 and Subs. 3.1 in arXiv:1210.2528. Subs. 3.1-3.2 (def. of an H-nice alg. and a formula for the Hopf PI-exp) coincide with Subs. 1.7-1.8 of arXiv:1207.1699. Lemmas 5 and 6 are adaptations of Lemmas 20 and 21 from arXiv:1207.1699 for a different cas

Theory of tangential idealizers and tangentially free ideals

We generalize the theory of logarithmic derivations through a self-contained study of modules here dubbed tangential idealizers. We establish reflexiveness criteria for such modules, provided the ring is a factorial domain. As a main consequence, necessary e sufficient conditions for their freeness are derived and the class of tangentially free ideals is introduced, thus extending (algebraically) the theory of free divisors proposed by K. Saito around 30 years ago.Comment: This submission has been withdrawn as it is subsumed by (and divided into) the following published papers: "Cleto B. Miranda-Neto, Tangential idealizers and differential ideals, Collect. Math. 67, 311-328, 2016", and "Cleto B. Miranda-Neto, Free logarithmic derivation modules over factorial domains, Math. Res. Lett. 24(1), 153-172, 2017

Finiteness properties of affine difference algebraic groups

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.Comment: 38 pages. This version (v2) is a major reorganization of the first version (v1). Roughly, the first four sections of v2 correspond to the first four sections of v1. The 5th section of v2 corresponds to the 6th section of v1 and the 6th section of v2 corresponds to the 9th section of v1. The 7th section of v2 is new. The sections of v1 not contained in v2 will appear elsewher

Representing Lie algebras using approximations with nilpotent ideals

We prove a refinement of Ado's theorem for Lie algebras over an algebraically-closed field of characteristic zero. We first define what it means for a Lie algebra $L$ to be approximated with a nilpotent ideal, and we then use such an approximation to construct a faithful representation of $L$. The better the approximation, the smaller the degree of the representation will be. We obtain, in particular, explicit and combinatorial upper bounds for the minimal degree of a faithful $L$-representation. The proofs use the universal enveloping algebra of Poincar\'e-Birkhoff-Witt and the almost-algebraic hulls of Auslander and Brezin

Effective Differential Nullstellensatz for Ordinary DAE Systems with Constant Coefficients

We give upper bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over any field of constants $K$ of characteristic $0$. Let $\vec{x}$ be a set of $n$ differential variables, $\vec{f}$ a finite family of differential polynomials in the ring $K\{\vec{x}\}$ and $f\in K\{\vec{x}\}$ another polynomial which vanishes at every solution of the differential equation system $\vec{f}=0$ in any differentially closed field containing $K$. Let $d:=\max\{\deg(\vec{f}), \deg(f)\}$ and $\epsilon:=\max\{2,{\rm{ord}}(\vec{f}), {\rm{ord}}(f)\}$. We show that $f^M$ belongs to the algebraic ideal generated by the successive derivatives of $\vec{f}$ of order at most $L = (n\epsilon d)^{2^{c(n\epsilon)^3}}$, for a suitable universal constant $c>0$, and $M=d^{n(\epsilon +L+1)}$. The previously known bounds for $L$ and $M$ are not elementary recursive

Combinatorics of binomial primary decomposition

An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.Comment: This paper was split off from math.AG/0610353 whose version 3 is now shorte

The model companion of differential fields with free operators

A model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Preprint 2013]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel's Lemma for local finite algebras over differentially closed fields

Abstract homomorphisms of algebraic groups and applications

This paper is an overview of my recent work on abstract homomorphisms of algebraic groups. It is based on a talk given at the Conference on Group Actions and Applications in Geometry, Topology, and Analysis held in Kunming in July 2012.Comment: arXiv admin note: substantial text overlap with arXiv:1005.0422, arXiv:1111.629

Quantifying singularities with differential operators

The $F$-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong $F$-regularity. However, it is very difficult to compute. Motivated by different aspects of the $F$-signature, we define a numerical invariant for rings of characteristic zero or $p>0$ that exhibits many of the useful properties of the $F$-signature. We also compute many examples of this invariant, including cases where the $F$-signature is not known. We also obtain a number of results on symbolic powers and Bernstein-Sato polynomials.Comment: 76 pages. Comments welcom