54 research outputs found
Quantitative equidistribution for the solutions of systems of sparse polynomial equations
For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},...,
x_n^{\pm1}] whose coefficients are not too big with respect to its directional
resultants, we show that the solutions in the algebraic n-th dimensional
complex torus of the system of equations f_1=\dots=f_n=0, are approximately
equidistributed near the unit polycircle. This generalizes to the multivariate
case a classical result due to Erdos and Turan on the distribution of the
arguments of the roots of a univariate polynomial. We apply this result to
bound the number of real roots of a system of Laurent polynomials, and to study
the asymptotic distribution of the roots of systems of Laurent polynomials with
integer coefficients, and of random systems of Laurent polynomials with complex
coefficients.Comment: 29 pages, 2 figures. Revised version, accepted for publication in the
American Journal of Mathematic
Arithmetic geometry of toric varieties. Metrics, measures and heights
We show that the height of a toric variety with respect to a toric metrized
line bundle can be expressed as the integral over a polytope of a certain
adelic family of concave functions. To state and prove this result, we study
the Arakelov geometry of toric varieties. In particular, we consider models
over a discrete valuation ring, metrized line bundles, and their associated
measures and heights. We show that these notions can be translated in terms of
convex analysis, and are closely related to objects like polyhedral complexes,
concave functions, real Monge-Amp\`ere measures, and Legendre-Fenchel duality.
We also present a closed formula for the integral over a polytope of a function
of one variable composed with a linear form. This allows us to compute the
height of toric varieties with respect to some interesting metrics arising from
polytopes. We also compute the height of toric projective curves with respect
to the Fubini-Study metric, and of some toric bundles.Comment: Revised version, 230 pages, 3 figure
Sobre corbes paramètriques i polÃgons de Newton
Les corbes i superfÃcies algebraiques poden ser definides implÃcitament com
a solucions d'equacions polinomials i, de vegades, també poden definir-se paramètricament,
mitjançant funcions racionals. Plantegem el problema de la conversió d'una
d'aquestes formes de representació a l'altra. A continuació, explorem la possibilitat
d'obtenir, a partir de les equacions paramètriques i sense necessitat d'efectuar l'operació
costosa de la implicitació, un objecte pròxim a les equacions implÃcites associades:
el polÃtop de Newton d'una hipersuperfÃcie donada paramètricament.Algebraic curves and surfaces can be defined as solutions of polynomial equations
and, sometimes, by parametric equations of rational functions as well. We
consider the problem of moving from parametric to implicit representations, a
usually involved process. We also explore the possibility of obtaining an object
close to the implicit equations from the parametric ones: the Newton polytop
of a hypersurface given in parametric form
A Sparse Effective Nullstellensatz
We present bounds for the sparseness in the Nullstellensatz. These bounds can give a much sharper characterization than degree bounds of the monomial structure of the polynomials in the Nullstellensatz in case that the input system is sparse. As a consequence we derive a degree bound which can substantially improve the known ones in case of a sparse system.In addition we introduce the notion of algebraic degree associated to a polynomial system of equations. We obtain a new degree bound which is sharper than the known ones when this parameter is small. We also improve the previous effective Nullstellensatze in case the input polynomials are quadratic.Our approach is completely algebraic, and the obtained results are independent of the characteristic of the base field.Facultad de Ciencias Exacta
The Canny–Emiris Conjecture for the Sparse Resultant
We present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.Fil: D'Andrea, Carlos. Centre de Recerca Matemà tica; España. Universidad de Barcelona; EspañaFil: Jeronimo, Gabriela Tali. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Sombra, MartÃn. Centre de Recerca Matemà tica; España. Institució Catalana de Recerca I Estudis Avançats; España. Universidad de Barcelona; Españ
Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze
Nous présentons des bornes pour les degrés et hauteurs des polynômes apparaissant dans certains problèmes de géométrie algébrique effective, dont l'implicitation d'applications rationnelles et le Nullstellensatz effectif sur une variété. Notre traitement est basé sur la théorie de l'intersection arithmétique dans un produit d'espaces projectifs. Il étend au cadre arithmétique des constructions et résultats dus à Jelonek. Un rôle central est joué par la notion de hauteur canonique mixte d'une variété multiprojective. Nous étudions cette notion à l'aide de la théorie des résultants et nous montrons quelques-unes de ses propriétés de base, y compris son comportement par rapport aux intersections, projections et produits. Nous obtenons aussi des résultats analogues dans le cas d'un corps de fonctions, dont un Nullstellensatz paramétrique.We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization 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.Fil: D'andrea, Carlos. Universidad de Barcelona; EspañaFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Sombra, MartÃn. Universidad de Barcelona; Españ
La géometrie du lieu flex d'une hypersurface
International audienceWe 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 in P3. 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
- …