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Korben Rusek

Publications and source records attributed to Korben Rusek.

8 recordsLinked to original sources
arXiv2017

A-Discriminants for Complex Exponents, and Counting Real Isotopy Types

We extend the definition of $\mathcal{A}$-discriminant varieties, and Kapranov's parametrization of $\mathcal{A}$-discriminant varieties, to complex exponents. As an application, we study the special case where $\mathcal{A}$ is a fixed real $n\times (n+3)$ matrix whose columns form the spectrum of an $n$-variate exponential sum $g$ with fixed sign vector for its coefficients: We prove that the number of possible isotopy types for the real zero set of $g$ is $O(n^2)$. The best previous upper bound was $2^{O(n^4)}$. Along the way, we also show that the singular loci of our generalized $\mathcal{A}$-discriminants are images of low-degree algebraic sets under certain analytic maps.

math.AG↗
arXiv2012

Effective Results on non-Archimedean Tropical Discriminants

We study A-discriminants from a non-Archimedean point of view, refining earlier work on the tropical discriminant. In particular, we study the case where $A$ is a collection of n+m+1 points in Z^n in general position, and give an algorithm to compute the image of the A-discriminant variety under the non-Archimedean evaluation map. When m=2, our approach yields tight lower and upper bounds, of order quadratic in n. We also detail a Sage package for plotting certain p-adic discriminant amoebae, and present explicit examples of point sets yielding discriminant amoebae with extremal behavior.

math.AG↗
arXiv2012

Report on "Mathematical Aspects of P vs. NP and its Variants."

This is a report on a workshop held August 1 to August 5, 2011 at the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University, Providence, Rhode Island, organized by Saugata Basu, Joseph M. Landsberg, and J. Maurice Rojas. We provide overviews of the more recent results presented at the workshop, including some works-in-progress as well as tentative and intriguing ideas for new directions. The main themes we discuss are representation theory and geometry in the Mulmuley-Sohoni Geometric Complexity Theory Program, and number theory and other ideas in the Blum-Shub-Smale model.

cs.CC↗
arXiv2010

Faster p-adic Feasibility for Certain Multivariate Sparse Polynomials

We present algorithms revealing new families of polynomials allowing sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we show that the case of honest n-variate (n+1)-nomials is doable in NP and, for p exceeding the Newton polytope volume and not dividing any coefficient, in constant time. Furthermore, using the theory of linear forms in p-adic logarithms, we prove that the case of trinomials in one variable can be done in NP. The best previous complexity bounds for these problems were EXPTIME or worse. Finally, we prove that detecting p-adic rational roots for sparse polynomials in one variable is NP-hard with respect to randomized reductions. The last proof makes use of an efficient construction of primes in certain arithmetic progressions. The smallest n where detecting p-adic rational roots for n-variate sparse polynomials is NP-hard appears to have been unknown.

math.NT↗
arXiv2010

Dense Fewnomials

We derive new bounds of fewnomial type for the number of real solutions to systems of polynomials that have structure intermediate between fewnomials and generic (dense) polynomials. This uses a modified version of Gale duality for polynomial systems. We also use stratified Morse theory to bound the total Betti number of a hypersurface defined by such a dense fewnomial. These bounds contain and generalize previous bounds for ordinary fewnomials obtained by Bates, Bertrand, Bihan, and Sottile.

math.AG↗
arXiv2010

Near NP-Completeness for Detecting p-adic Rational Roots in One Variable

We show that deciding whether a sparse univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic p-adic Newton polygon. We thus improve the best previous complexity upper bound of EXPTIME. We also prove an unconditional complexity lower bound of NP-hardness with respect to randomized reductions for general univariate polynomials. The best previous lower bound assumed an unproved hypothesis on the distribution of primes in arithmetic progression. We also discuss how our results complement analogous results over the real numbers.

math.NT↗
arXiv2007

Extremal Real Algebraic Geometry and A-Discriminants

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recent formula for the A-discriminant, and give new bounds on the topology of certain A-discriminant varieties. A consequence of the latter result is a new upper bound on the number of topological types of certain real algebraic sets defined by sparse polynomial equations, e.g., the number of smooth topological types attainable in certain families of real algebraic surfaces.

math.AG↗