Related papers: Lower bounds on the essential dimension of reducti…

We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the…

In this paper we study the scalar geometries occurring in the dimensional reduction of minimal five-dimensional supergravity to three Euclidean dimensions, and find that these depend on whether one first reduces over space or over time. In…

We prove a sharp bound for the minimal doubling of a small measurable subset of a compact connected Lie group. Namely, let $G$ be a compact connected Lie group of dimension $d_G$, we show that for for all measurable subsets $A$, we have…

We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new…

Parahoric group schemes are certain possibly non-reductive, smooth, affine integral models of reductive group schemes defined over a henselian discretely valued field $K$ whose residue field is perfect. We show that any such group scheme…

The minimum distance of expander codes over GF(q) is studied. A new upper bound on the minimum distance of expander codes is derived. The bound is shown to lie under the Varshamov-Gilbert (VG) bound while q >= 32. Lower bounds on the…

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do…

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives…

Due to their importance in both data analysis and numerical algorithms, low rank approximations have recently been widely studied. They enable the handling of very large matrices. Tight error bounds for the computationally efficient…

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the…

Recently, Korkmaz established the lower bound of $3g - 2$ for the dimension of a faithful representation of the mapping class group of an orientable surface of genus $g \ge 3$. We raise this bound to $4g - 3$ in the setting of surfaces of…

Given a pseudo-effective divisor L we construct the diminished ideal of L, a "continuous" extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. For most pseudo-effective divisors L the multiplier…

A fundamental component of neural network verification is the computation of bounds on the values their outputs can take. Previous methods have either used off-the-shelf solvers, discarding the problem structure, or relaxed the problem even…

Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard problem for general linear codes. In practice, one often uses codes with additional…

We give near-tight lower bounds for the sparsity required in several dimensionality reducing linear maps. First, consider the JL lemma which states that for any set of n vectors in R there is a matrix A in R^{m x d} with m = O(eps^{-2}log…

We give a simple formula for the essential dimension of a finite pseudo-reflection group at a prime p and determine the absolute essential dimension for most irreducible pseudo-reflection groups. We also study the "poor man's essential…

Let $k$ be an algebraically closed field of characteristic $p>0$ and $G$ a finite group. We provide a description of the torsion subgroup $TT(G)$ of the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules when $p=2$ and $G$…

We summarize what is currently known about ed$(G)$ and RD$(G)$ for finite groups $G$ over $\mathbb{C}$ (i.e. in characteristic 0). In Appendix A, we also give an argument which improves the known bound on RD(PSL$(2,\mathbb{F}_{11})$).

Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition…

This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial…