Related papers: Newton Binomial Formulas in Schubert Calculus
In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it.
We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns 1432 and 1423 is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about…
We use Newton divided differences for calculation of Greene sums -- the rational functions determined by linear extensions of partially ordered sets. Identities for Greene sums generate relations for Newton divided differences and Arnold…
Using a sums of squares formula for two variable polynomials with no zeros on the bidisk, we are able to give a new proof of a representation for distinguished varieties. For distinguished varieties with no singularities on the two-torus,…
We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…
The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum…
We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm…
We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface…
A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection…
We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real.
We try to understand and justify Schubert Calculus the way Schubert did it.
We generalize Sylvester single sums to multisets (sets with repeated elements), and show that these sums compute subresultants of two univariate polyomials as a function of their roots independently of their multiplicity structure. This is…
We establish necessary and sufficient conditions for a polynomial to be divisible by a cyclotomic polynomials and derive new formulas involving Ramanujan sums as an application of our results. Additionally, we provide new insights into the…
We investigate Newton series for truncated multiple $L$-values and thereby obtain a class of relations for multiple $L$-values. In addition, we give a formulation and a proof of extended derivation relations for multiple $L$-values.
We establish a Schubert calculus for Bott-Samelson resolutions in the algebraic cobordism ring of a complete flag variety G/B.
The Newton-Girard Formula allows one to write any elementary symmetric polynomial as a sum of products of power sum symmetric polynomials and elementary symmetric polynomials of lesser degree. It has numerous applications. We have…
We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic…
We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture, and describe completely the solution set.
This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.
In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging…