Quantifier Elimination over Finite Fields Using Gr\"obner Bases

We give an algebraic quantifier elimination algorithm for the first-order theory over any given finite field using Gr\"obner basis methods. The algorithm relies on the strong Nullstellensatz and properties of elimination ideals over finite fields. We analyze the theoretical complexity of the algorithm and show its application in the formal analysis of a biological controller model.Comment: A shorter version is to appear in International Conference on Algebraic Informatics 201

Invariants of singularities of pairs

Let X be a smooth complex variety and Y be a closed subvariety of X, or more generally, a closed subscheme of X. We are interested in invariants attached to the singularities of the pair (X, Y). We discuss various methods to construct such invariants, coming from the theory of multiplier ideals, D-modules, the geometry of the space of arcs and characteristic p techniques. We present several applications of these invariants to algebra, higher dimensional birational geometry and to singularities.Comment: 19 pages, to appear in Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 (talk to be given by the first author); some typos correcte

Algebraic method for finding equivalence groups

The algebraic method for computing the complete point symmetry group of a system of differential equations is extended to finding the complete equivalence group of a class of such systems. The extended method uses the knowledge of the corresponding equivalence algebra. Two versions of the method are presented, where the first involves the automorphism group of this algebra and the second is based on a list of its megaideals. We illustrate the megaideal-based version of the method with the computation of the complete equivalence group of a class of nonlinear wave equations with applications in nonlinear elasticity.Comment: 17 pages; revised version; includes results that have been excluded from the journal version of the preprint arXiv:1106.4801v

Cohomology on Toric Varieties and Local Cohomology with Monomial Supports

We study the local cohomology modules H^i_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X_1,...,X_n]. We consider a grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional and we give an effective way to compute these components. Using Cox's description for sheaves on toric varieties, we apply these results to compute the cohomology of coherent sheaves on toric varieties. We give algorithms for this computation which have been implemented in the Macaulay 2 system. We obtain also a topological description for the cohomology of rank one torsionfree sheaves on toric varieties.Comment: 23 pages, 2 figures, uses diagrams.tex, to appear in Journal of Symbolic Computatio

A decision method for the integrability of differential-algebraic Pfaffian systems

We prove an effective integrability criterion for differential-algebraic Pfaffian systems leading to a decision method of consistency with a triple exponential complexity bound. As a byproduct, we obtain an upper bound for the order of differentiations in the differential Nullstellensatz for these systems

Rational, Replacement, and Local Invariants of a Group Action

The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame method in differential geometry. The generating set of rational invariants appears as the coefficients of a Groebner basis, reduction with respect to which allows to express a rational invariant in terms of the generators. The replacement invariants, introduced in the paper, are tuples of algebraic functions of the rational invariants. Any invariant, whether rational, algebraic or local, can be can be rewritten terms of replacement invariants by a simple substitution.Comment: 37 page

Lyubeznik numbers of local rings and linear strands of graded ideals

In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. We also prove that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. In particular, they satisfy a consecutiveness property that we prove first for Lyubeznik numbers. For the case of squarefree monomial ideals we get more insight on the relation between Lyubeznik numbers and the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik numbers of Stanley-Reisner rings are not only an algebraic invariant but also a topological invariant, meaning that they depend on the homeomorphic class of the geometric realization of the associated simplicial complex and the characteristic of the base field.Comment: 25 pages. Accepted in Nagoya Math.

Strong FF-regularity and generating morphisms of local cohomology modules

We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an $n\times (n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of $X$ is strongly $F$-regular.Comment: 18 page

The Complexity of the Ideal Membership Problem for Constrained Problems Over the Boolean Domain

Given an ideal $I$ and a polynomial $f$ the Ideal Membership Problem is to test if $f\in I$. This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the Ideal Membership Problem for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the borderline of tractability. By using Gr\"{o}bner bases techniques, we extend Schaefer's dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems over the Boolean domain to be either in P or NP-hard. Moreover, our result implies necessary and sufficient conditions for the efficient computation of Theta Body SDP relaxations, identifying therefore the borderline of tractability for constraint language problems. This paper is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares proofs [O'Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the Ideal Membership Problem tractability for combinatorial ideals implies bounded coefficients in Sum-of-Squares proofs.Comment: Preliminary version appeared in ACM-SIAM Symposium on Discrete Algorithms (SODA19

Block-Krylov techniques in the context of sparse-FGLM algorithms

Consider a zero-dimensional ideal $I$ in $\mathbb{K}[X_1,\dots,X_n]$. Inspired by Faug\`ere and Mou's Sparse FGLM algorithm, we use Krylov sequences based on multiplication matrices of $I$ in order to compute a description of its zero set by means of univariate polynomials. Steel recently showed how to use Coppersmith's block-Wiedemann algorithm in this context; he describes an algorithm that can be easily parallelized, but only computes parts of the output in this manner. Using generating series expressions going back to work of Bostan, Salvy, and Schost, we show how to compute the entire output for a small overhead, without making any assumption on the ideal $I$ other than it having dimension zero. We then propose a refinement of this idea that partially avoids the introduction of a generic linear form. We comment on experimental results obtained by an implementation based on the C++ libraries Eigen, LinBox and NTL.Comment: 32 pages, 7 algorithms, 2 table