Teresa Krick

Sharp estimates for the arithmetic Nullstellensatz

We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring ZZ . The result improves previous work of Philippon, Berenstein-Yger and Krick-Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application, we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine variety defined over a number field. We introduce this notion and study its basic properties.

A computational method for diophantine approximation

The procedures to solve algebraic geometry elimination problems have usually been designed from the point of view of commutative algebra. For instance, let us consider the problem of deciding whether a given system of polynomial equalities has a solution. This means that we have to eliminate a single block of quantifiers in a formula with polynomial equations.

Deformation techniques for efficient polynomial equation solving

Abstract Suppose we are given a parametric polynomial equation system encoded by an arithmetic circuit, which represents a generically flat and unramified family of zero-dimensional algebraic varieties. Let us also assume that there is given the complete description of the solution of a particular unramified parameter instance of our system. We show that it is possible to “move” the given particular solution along the parameter space in order to reconstruct—by means of an arithmetic circuit—the coordinates of the solutions of the system for an arbitrary parameter instance. The underlying algorithm is highly efficient, i.e., polynomial in the syntactic description of the input and the following geometric invariants: the number of solutions of a typical parameter instance and the degree of the polynomials occurring in the output. In fact, we prove a slightly more general result, which implies the previous statement by means of a well-known primitive element algorithm. We produce an efficient algorithmic description of the hypersurface obtained projecting polynomially the given generically flat family of varieties into a suitable affine space.

The Computational Complexity of the Chow Form

Abstract We present a bounded probability algorithm for the computation of the Chowforms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system.

Factoring bivariate sparse (lacunary) polynomials

Abstract We present a deterministic algorithm for computing all irreducible factors of degree ⩽ d of a given bivariate polynomial f ∈ K [ x , y ] over an algebraic number field K and their multiplicities, whose running time is polynomial over the rationals, in the bit length of the sparse encoding of the input and in d. Moreover, we show that the factors over Q ¯ of degree ⩽ d which are not binomials can also be computed in time polynomial in the sparse length of the input and in d.

Membership problem, Representation problem and the Computation of the Radical for one-dimensional Ideals

In this paper we consider membership problem, representation problem and also the computation of the radical for one-dimensional ideals in the polynomial ring k[X 1,…, X n ] from a complexity point of view. Our aim is to give bounds for the complexity of the above problems which are simply exponential in the number n of variables in the one-dimensional case. Moreover we show that in the general case the first two problems are doubly exponential only in the dimension of the ideal, and parallelizable. Many authors considered membership problem (i.e. given polynomials F, F 1,…, F, ∈ k[X 1,…, X n ], decide whether F belongs to I:= (F1…, F,)) and representation problem (compute a representation A i ∈ k[X 1,…, X n ]) from this effective approach. In particular [18] gives a lower bound, doubly exponential in (a fraction of) the numbers of variables n. On the other hand, [4] shows that in “good” cases, for instance for unmixed ideals, the membership problem is simply exponential in n.

Algorithmic aspects of Suslin's proof of Serre's conjecture

AbstractLetF be a unimodularr×s matrix with entries beingn-variate polynomials over an infinite fieldK. Denote by deg(F) the maximum of the degrees of the entries ofF and letd=1+deg(F). We describe an algorithm which computes a unimodulars×s matrixM with deg(M)=(rd)O(n) such thatFM=[Ir,O], where [Ir,O] denotes ther×s matrix obtained by adding to ther×r unit matrixIrs−r zero columns.We present the algorithm as an arithmetic network with inputs fromK, and we count field operations and comparisons as unit cost.The sequential complexity of our algorithm amounts to

A numerical algorithm for zero counting. II: Distance to ill-posedness and smoothed analysis

S^{O(r^2 )} r^{O(n^2 )} d^{O(n^2 + r^2 )}