arXiv subjects
Publications and source records attributed to Martin Sombra.
We present some results on projective toric varieties which are relevant in Diophantine geometry. We interpret and study several invariants attached to these varieties in geometrical and combinatorial terms. We also give a Bézout theorem for the Chow weights of projective varieties and an application to the theorem of successive minima.
We present an explicit expression for the normalized height of a projective toric variety. This expression decomposes as a sum of local contributions, each term being the integral of a certain function, concave and piecewise linear-affine. More generally, we obtain an explicit expression for the normalized multiheight of a torus with respect to several monomial embeddings. The set of functions introduced behaves as an arithmetic analog of the polytope classically associated with the torus action. Besides the formulae for the height and multiheight, we show that this object behaves in a natural way with respect to several standard constructions: decomposition into orbits, joins, Segre products and Veronese embeddings. The proof follows an indirect way: instead of the definition of the normalized height, we rely on the computation of an appropriate arithmetic Hilbert function.
We prove that the Cayley-Menger determinant of an $n$-dimensional simplex is an absolutely irreducible polynomial for $n\geq3.$ We also study the irreducibility of polynomials associated to related geometric constructions.
We compute the successive minima of the projective toric variety $X_\cA$ associated to a finite set $ \cA \subset \Z^n$. As a consequence of this computation and of the results of S.-W. Zhang on the distribution of small points, we derive estimates for the height of the subvariety $X_\cA$ and of the $\cA$-resultant. These estimates allow us to obtain an arithmetic analogue of the Bezout-Kushnirenko's theorem concerning the number of solutions of a system of polynomial equations. As an application of this result, we improve the known estimates for the height of the polynomials in the sparse Nullstellensatz.
We present an upper bound for the height of the mixed sparse resultant, defined as the logarithm of the maximum modulus of its coefficients. We obtain a similar estimate for its Mahler measure.
We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system defining the variety. In particular, it provides an alternative algorithm for the equidimensional decomposition of a variety. As an application we obtain an algorithm for the computation of a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As a further application, we derive an algorithm for the computation of the (unique) solution of a generic over-determined equation system.
We introduce a notion of palindromicity of a natural number which is independent of the base. We study the existence and density of palindromic and multiple palindromic numbers, and we raise several related questions.
We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over $\Z$. 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.
We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the {\it geometric degree of the system of equations}. The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases. The proof of this result is combinatorial, and it relies on global estimations for the Hilbert function of homogeneous polynomial ideals. In this direction, we obtain a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper bound for the Hilbert function of a generic hypersurface section of an unmixed radical polynomial ideal.