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Martin Sombra

Publications and source records attributed to Martin Sombra.

At least 19 recordsLinked to original sources
arXiv2022

Arithmetic positivity on toric varieties

We continue with our study of the arithmetic geometry of toric varieties. In this text, we study the positivity properties of metrized R-divisors in the toric setting. For a toric metrized R-divisor, we give formulae for its arithmetic volume and its chi-arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data. As an application, we prove a Dirichlet's unit theorem on toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized R-divisor, and we prove a toric arithmetic Fujita approximation theorem.

math.AG↗
arXiv2021

The Canny-Emiris conjecture for the sparse resultant

We present a product formula for the initial parts of the sparse resultant associated to an arbitrary family of supports, generalising a previous result by Sturmfels. This allows to compute the homogeneities and degrees of the sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain a similar product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated to a mixed subdivision of a polytope. Applying these results, we prove that the sparse resultant can be computed as the quotient of the determinant of such a square matrix by a certain principal minor, under suitable hypothesis. This generalises the classical Macaulay formula for the homogeneous resultant, and confirms a conjecture of Canny and Emiris.

math.AC↗
arXiv2021

Bounds for multivariate residues and for the polynomials in the elimination theorem

We present several upper bounds for the height of global residues of rational forms on an affine variety. As a consequence, we deduce upper bounds for the height of the coefficients in the Bergman-Weil trace formula. We also present upper bounds for the degree and the height of the polynomials in the elimination theorem on an affine variety. This is an arithmetic analogue of Jelonek's effective elimination theorem, that plays a crucial role in the proof of our bounds for the height of global residues.

math.CV↗
arXiv2019

The geometry of the flex locus of a hypersurface

We give a formula in terms of multidimensional resultants for an equation for the flex locus of a projective hypersurface, generalizing a classical result of Salmon for surfaces. Using this formula, we compute the dimension of this flex locus, and an upper bound for the degree of its defining equations. We also show that, when the hypersurface is generic, this bound is reached, and that the generic flex line is unique and has the expected order of contact with the hypersurface.

math.AG↗
arXiv2017

Reductions Modulo Primes of Systems of Polynomial Equations and Algebraic Dynamical Systems

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the algebraic closure of the field with $p$ elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture.

math.NT↗
arXiv2017

Orbits of Polynomial Dynamical Systems Modulo Primes

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and Ye~(2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang~(2015). As a by-product, we also slighly improve a result of Silverman~(2008) and recover a result of Akbary and Ghioca~(2009) as special extreme cases of our estimates.

math.NT↗
arXiv2015

Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions

We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matoušek, Sharir and Safernová. We also present a bound for the number of incidences between points and hypersurfaces in the four-dimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components of a semi-algebraic set by Barone and Basu.

math.AG↗
arXiv2015

Successive minima of toric height functions

Given a toric metrized R-divisor on a toric variety over a global field, we give a formula for the essential minimum of the associated height function. Under suitable positivity conditions, we also give formulae for all the successive minima. We apply these results to the study, in the toric setting, of the relation between the successive minima and other arithmetic invariants like the height and the arithmetic volume. We also apply our formulae to compute the successive minima for several families of examples, including weighted projective spaces, toric bundles and translates of subtori.

math.NT↗
arXiv2014

A Poisson formula for the sparse resultant

We present a Poisson formula for sparse resultants and a formula for the product of the roots of a family of Laurent polynomials, which are valid for arbitrary families of supports. To obtain these formulae, we show that the sparse resultant associated to a family of supports can be identified with the resultant of a suitable multiprojective toric cycle in the sense of Remond. This connection allows to study sparse resultants using multiprojective elimination theory and intersection theory of toric varieties.

math.AG↗
arXiv2014

Height of varieties over finitely generated fields

We show that the height of a variety over a finitely generated field of characteristic zero can be written as an integral of local heights over the set of places of the field. This allows us to apply our previous work on toric varieties and extend our combinatorial formulae for the height to compute some arithmetic intersection numbers of non toric arithmetic varieties over the rational numbers.

math.NT↗
arXiv2014

Overdetermined systems of sparse polynomial equations

We show that, for a system of univariate polynomials given in sparse encoding, we can compute a single polynomial defining the same zero set, in time quasi-linear in the logarithm of the degree. In particular, it is possible to determine whether such a system of polynomials does have a zero in time quasi-linear in the logarithm of the degree. The underlying algorithm relies on a result of Bombieri and Zannier on multiplicatively dependent points in subvarieties of an algebraic torus. We also present the following conditional partial extension to the higher dimensional setting. Assume that the effective Zilber conjecture holds. Then, for a system of multivariate polynomials given in sparse encoding, we can compute a finite collection of complete intersections outside hypersurfaces that defines the same zero set, in time quasi-linear in the logarithm of the degree.

math.AG↗
arXiv2012

Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze

We present bounds for the degree and the height of the polynomials arising in some central problems in effective algebraic geometry including the implicitation of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz.

math.AG↗
arXiv2010

The Newton polygon of a rational plane curve

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kushnirenko-Bernstein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic rational functions, with explicit genericity conditions. We also show that the variety of rational curves with given Newton polygon is unirational and we compute its dimension. As a consequence, we obtain that any convex lattice polygon with positive area is the Newton polygon of a rational plane curve.

math.AG↗
arXiv2007

A refinement of the Kushnirenko-Bernstein estimate

A theorem of Kushnirenko and Bernstein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.

math.AG↗
arXiv2006

Essential minimum and obstruction degrees of translates of subtori

We study the obstruction degrees of translates of sub-tori of multiplicative tori and we show how they are connected to lower bounds for the essential minimum of these varieties. In particular, we combine our computations with results of A. Schinzel and of F. Amoroso - R. Dvornicich, that we thus extend to translates of sub-tori defined over CM fields. This shows that the essential minimum of these varieties have a geometric behavior similar to that of varieties which are not translates of sub-tori. We also refine and extend to translates of sub-tori a lower bound obtained by D. Bertrand in connexion with relations of multiplicative dependence of algebraic points. Finaly, we prove that in the functional case the conjectural lower bound for the essential minimum of sub-varieties of multiplicative tori is true and its proof is elementary.

math.NT↗
arXiv2006

Quelques aspects diophantiens des varietes toriques projectives

Some diophantine aspects of projective toric varieties: We present several faces of projective toric varieties, of interest from the point of view of diophantine geometry. We make explicit the theory on a number of meaningful examples and we also prove a Bezout type theorem for Chow weight of projective varieties.

math.NT↗
arXiv2006

Factoring bivariate sparse (lacunary) polynomials

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

math.NT↗