Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials

International audienceWe use the modules introduced by Kraśkiewicz and Pragacz (1987, 2004) to show some positivity propertiesof Schubert polynomials. We give a new proof to the classical fact that the product of two Schubert polynomialsis Schubert-positive, and also show a new result that the plethystic composition of a Schur function with a Schubertpolynomial is Schubert-positive. The present submission is an extended abstract on these results and the full versionof this work will be published elsewhere.Nous employons les modules introduits par Kraśkiewicz et Pragacz (1987, 2004) et démontrons certainespropriétés de positivité des polynômes de Schubert: nous donnons une nouvelle preuve pour le fait classique quele produit de deux polynômes de Schubert est Schubert-positif; nous démontrons aussi un nouveau résultat que lacomposition plethystique d’une fonction de Schur avec un polynôme de Schubert est Schubert-positif. Cet article estun sommaire de ces résultats, et une version pleine de ce travail sera publée ailleurs

Linear Programming Relaxations for Goldreich's Generators over Non-Binary Alphabets

Goldreich suggested candidates of one-way functions and pseudorandom generators included in $\mathsf{NC}^0$. It is known that randomly generated Goldreich's generator using $(r-1)$-wise independent predicates with $n$ input variables and $m=C n^{r/2}$ output variables is not pseudorandom generator with high probability for sufficiently large constant $C$. Most of the previous works assume that the alphabet is binary and use techniques available only for the binary alphabet. In this paper, we deal with non-binary generalization of Goldreich's generator and derives the tight threshold for linear programming relaxation attack using local marginal polytope for randomly generated Goldreich's generators. We assume that $u(n)\in \omega(1)\cap o(n)$ input variables are known. In that case, we show that when $r\ge 3$, there is an exact threshold $\mu_\mathrm{c}(k,r):=\binom{k}{r}^{-1}\frac{(r-2)^{r-2}}{r(r-1)^{r-1}}$ such that for $m=\mu\frac{n^{r-1}}{u(n)^{r-2}}$, the LP relaxation can determine linearly many input variables of Goldreich's generator if $\mu>\mu_\mathrm{c}(k,r)$, and that the LP relaxation cannot determine $\frac1{r-2} u(n)$ input variables of Goldreich's generator if $\mu<\mu_\mathrm{c}(k,r)$. This paper uses characterization of LP solutions by combinatorial structures called stopping sets on a bipartite graph, which is related to a simple algorithm called peeling algorithm.Comment: 14 pages, 1 figur