Symmetric polynomials and divided differences in formulas of intersection theory

The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci; some explicit formulas for the Chern and Segre classes of Schur bundles with applications to enumerative geometry; flag degeneracy loci; fundamental classes, diagonals and Gysin maps; intersection rings of G/P and formulas for isotropic degeneracy loci; numerically positive polynomials for ample vector bundles. Apart of surveyed results, the paper contains also some new results as well as some new proofs of earlier ones: how to compute the fundamental class of a subvariety from the class of the diagonal of the ambient space; how to compute the class of the relative diagonal using Gysin maps; a new formula for pushing forward Schur's Q- polynomials in Grassmannian bundles; a new formula for the total Chern class of a Schur bundle; another proof of Schubert's and Giambelli's enumeration of complete quadrics; an operator proof of the Jacobi-Trudi formula; a Schur complex proof of the Giambelli-Thom-Porteous formula.Comment: 58 pages; to appear in the volume "Parameter Spaces", Banach Center Publications vol 36 (1996) AMSTE

Integer-valued polynomials over matrices and divided differences

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$, the $k$-th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$. If in addition the intersection of the maximal ideals of finite index is $(0)$ then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$, that is, conjugate integral elements of degree $n$ over $D$.Comment: minor changes, notation made uniform throughout the paper. Fixed a wrong assumption we used in (4), (5) and Thm 4.1: "$D$ has zero Jacobson radical" has to be replaced with "the intersection of the maximal ideals of finite index is $(0)$". Keywords: Integer-valued polynomial, Divided differences, Matrix, Integral element, Polynomial closure, Pullback. In Monatshefte f\"ur Mathematik, 201

Thom polynomials and Schur functions: the singularities I_{2,2}(-)

We give the Thom polynomials for the singularities $I_{2,2}$ associated with maps $({\bf C}^{\bullet},0) \to ({\bf C}^{\bullet+k},0)$ with parameter $k\ge 0$. Our computations combine the characterization of Thom polynomials via the ``method of restriction equations'' of Rimanyi et al. with the techniques of Schur functions.Comment: 21 pages; Ann. Inst. Fourier vol.57; this is expanded Sect.4 of math.AG/0509234; new added results: Theorem 11 (based on P-ideals of singularities) and explicit expressions for the coefficients of the Thom polynomials of I_22(-) (Propositions 17, 18, 19); references update

The ring of polynomials integral-valued over a finite set of integral elements

Let $D$ be an integral domain with quotient field $K$ and $\Omega$ a finite subset of $D$. McQuillan proved that the ring ${\rm Int}(\Omega,D)$ of polynomials in $K[X]$ which are integer-valued over $\Omega$, that is, $f\in K[X]$ such that $f(\Omega)\subset D$, is a Pr\"ufer domain if and only if $D$ is Pr\"ufer. Under the further assumption that $D$ is integrally closed, we generalize his result by considering a finite set $S$ of a $D$-algebra $A$ which is finitely generated and torsion-free as a $D$-module, and the ring ${\rm Int}_K(S,A)$ of integer-valued polynomials over $S$, that is, polynomials over $K$ whose image over $S$ is contained in $A$. We show that the integral closure of ${\rm Int}_K(S,A)$ is equal to the contraction to $K[X]$ of ${\rm Int}(\Omega_S,D_F)$, for some finite subset $\Omega_S$ of integral elements over $D$ contained in an algebraic closure $\bar K$ of $K$, where $D_F$ is the integral closure of $D$ in $F=K(\Omega_S)$. Moreover, the integral closure of ${\rm Int}_K(S,A)$ is Pr\"ufer if and only if $D$ is Pr\"ufer. The result is obtained by means of the study of pullbacks of the form $D[X]+p(X)K[X]$, where $p(X)$ is a monic non-constant polynomial over $D$: we prove that the integral closure of such a pullback is equal to the ring of polynomials over $K$ which are integral-valued over the set of roots $\Omega_p$ of $p(X)$ in $\bar K$.Comment: final version, J. Commut. Algebra 8 (2016), no. 1, 113-14

Moment ideals of local Dirac mixtures

In this paper we study ideals arising from moments of local Dirac measures and their mixtures. We provide generators for the case of first order local Diracs and explain how to obtain the moment ideal of the Pareto distribution from them. We then use elimination theory and Prony's method for parameter estimation of finite mixtures. Our results are showcased with applications in signal processing and statistics. We highlight the natural connections to algebraic statistics, combinatorics and applications in analysis throughout the paper.Comment: 26 pages, 3 figure

Subword complexes in Coxeter groups

Let (\Pi,\Sigma) be a Coxeter system. An ordered list of elements in \Sigma and an element in \Pi determine a {\em subword complex}, as introduced in our paper on Gr\"obner geometry of Schubert polynomials (math.AG/0110058). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which is due to Fomin and Kirillov, are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.Comment: 14 pages. Final version, to appear in Advances in Mathematics. This paper was split off from math.AG/0110058v2, whose version 3 is now shorte