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J. Maurice Rojas

Publications and source records attributed to J. Maurice Rojas.

At least 55 records · Page 3Linked to original sources
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↗
arXiv2006

A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes

We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: (*) Decide whether a univariate polynomial with exactly m monomial terms has a p-adic rational root. In particular, we show that while (*) is doable in quantum randomized polynomial time when m=2 (and no classical randomized polynomial time algorithm is known), (*) is nearly NP-hard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (*) would imply the widely disbelieved inclusion NP \subseteq BPP. This type of quantum/classical interpolation phenomenon appears to new.

quant-ph↗
arXiv2006

First Steps in Algorithmic Fewnomial Theory

Fewnomial theory began with explicit bounds -- solely in terms of the number of variables and monomial terms -- on the number of real roots of systems of polynomial equations. Here we take the next logical step of investigating the corresponding existence problem: Let FEAS_R denote the problem of deciding whether a given system of multivariate polynomial equations with integer coefficients has a real root or not. We describe a phase-transition for when m is large enough to make FEAS_R be NP-hard, when restricted to inputs consisting of a single n-variate polynomial with exactly m monomial terms: polynomial-time for m<=n+2 (for any fixed n) and NP-hardness for m<=n+n^{epsilon} (for n varying and any fixed epsilon>0). Because of important connections between FEAS_R and A-discriminants, we then study some new families of A-discriminants whose signs can be decided within polynomial-time. (A-discriminants contain all known resultants as special cases, and the latter objects are central in algorithmic algebraic geometry.) Baker's Theorem from diophantine approximation arises as a key tool. Along the way, we also derive new quantitative bounds on the real zero sets of n-variate (n+2)-nomials.

math.AG↗
arXiv2004

On Solving Fewnomials Over Intervals in Fewnomial Time

Let f be a degree D univariate polynomial with real coefficients and exactly m monomial terms. We show that in the special case m=3 we can approximate within eps all the roots of f in the interval [0,R] using just O(log(D)log(Dlog(R/eps))) arithmetic operations. In particular, we can count the number of roots in any bounded interval using just O(log^2 D) arithmetic operations. Our speed-ups are significant and near-optimal: The asymptotically sharpest previous complexity upper bounds for both problems were super-linear in D, while our algorithm has complexity close to the respective complexity lower bounds. We also discuss conditions under which our algorithms can be extended to general m, and a connection to a real analogue of Smale's 17th Problem.

math.NA↗
arXiv2003

A Direct Ultrametric Approach to Additive Complexity and the Shub-Smale Tau Conjecture

The Shub-Smale Tau Conjecture is a hypothesis relating the number of integral roots of a polynomial f in one variable and the Straight-Line Program (SLP) complexity of f. A consequence of the truth of this conjecture is that, for the Blum-Shub-Smale model over the complex numbers, P differs from NP. We prove two weak versions of the Tau Conjecture and in so doing show that the Tau Conjecture follows from an even more plausible hypothesis. Our results follow from a new p-adic analogue of earlier work relating real algebraic geometry to additive complexity. For instance, we can show that a nonzero univariate polynomial of additive complexity s can have no more than 15+s^3(s+1)(7.5)^s s! =O(e^{s\log s}) roots in the 2-adic rational numbers Q_2, thus dramatically improving an earlier result of the author. This immediately implies the same bound on the number of ordinary rational roots, whereas the best previous upper bound via earlier techniques from real algebraic geometry was a quantity in Omega((22.6)^{s^2}). This paper presents another step in the author's program of establishing an algorithmic arithmetic version of fewnomial theory.

math.NT↗
arXiv2003

Dedekind Zeta Functions and the Complexity of Hilbert's Nullstellensatz

Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN in P implies P=NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication that HN not in NP implies P is not equal to NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence on the underlying field, while the second can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also apply recent explicit algebraic and analytic estimates, some due to Silberman and Sombra, which may be of independent interest.

math.NT↗
arXiv2002

Why Polyhedra Matter in Non-Linear Equation Solving

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein's Theorem. Our extension relates volumes of polytopes with the number of connected components of the complex zero set of a polynomial system, and allows any number of polynomials and/or variables. (2) A near optimal complexity bound for computing mixed area -- a quantity intimately related to counting complex roots in the plane.

math.AG↗
arXiv2002

Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a single n-variate m-nomial.

math.AG↗
arXiv2002

High Probability Analysis of the Condition Number of Sparse Polynomial Systems

Let F:=(f_1,...,f_n) be a random polynomial system with fixed n-tuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/epsilon. The bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the Newton polytopes (and underlying covariances) are all identical. We also consider polynomials with real coefficients and give bounds for the expected number of real roots and (restricted) condition number. Using a Kahler geometric framework throughout, we also express the expected number of roots of f inside a region U as the integral over U of a certain {\bf mixed volume} form, thus recovering the classical mixed volume when U = (C^*)^n.

math.NA↗
arXiv2002

Additive Complexity and the Roots of Polynomials Over Number Fields and p-adic Fields

Consider any nonzero univariate polynomial with rational coefficients, presented as an elementary algebraic expression (using only integer exponents). Letting sigma(f) denotes the additive complexity of f, we show that the number of rational roots of f is no more than 15 + sigma(f)^2 (24.01)^{sigma(f)} sigma(f)!. This provides a sharper arithmetic analogue of earlier results of Dima Grigoriev and Jean-Jacques Risler, which gave a bound of C^{sigma(f)^2} for the number of real roots of f, for some constant C with 1<C<32. We extend our new bound to arbitrary finite extensions of the ordinary or p-adic rationals, roots of bounded degree over a number field, and geometrically isolated roots of multivariate polynomial systems. We thus extend earlier bounds of Hendrik W. Lenstra, Jr. and the author to encodings more efficient than monomial expansions. We also mention a connection to complexity theory and note that our bounds hold for a broader class of fields.

math.NT↗
arXiv2001

Arithmetic Multivariate Descartes' Rule

Let L be any number field or $\mathfrak{p}$-adic field and consider F:=(f_1,...,f_k) where f_i is in L[x_1,...,x_n]\{0} for all i and there are exactly m distinct exponent vectors appearing in f_1,...,f_k. We prove that F has no more than 1+(cmn(m-1)^2 log m)^n geometrically isolated roots in L^n, where c is an explicit and effectively computable constant depending only on L. This gives a significantly sharper arithmetic analogue of Khovanski's Theorem on Fewnomials and a higher-dimensional generalization of an earlier result of Hendrik W. Lenstra, Jr. for the case of a single univariate polynomial. We also present some further refinements of our new bounds and briefly discuss the complexity of finding isolated rational roots.

math.NT↗
arXiv2001

Counting Isolated Roots of Trinomial Systems in the Plane and Beyond

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive new bounds on the number of real connected components of fewnomial hypersurfaces.

math.CO↗
arXiv2001

Finiteness for Arithmetic Fewnomial Systems

Suppose L is any finite algebraic extension of either the ordinary rational numbers or the p-adic rational numbers. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms appearing in at least one g_i is exactly m. We prove that the maximum number of isolated roots of G:=(g_1,...,g_k) in L^n is finite and depends solely on (m,n,L), i.e., is independent of the degrees of the g_i. We thus obtain an arithmetic analogue of Khovanski's Theorem on Fewnomials, extending earlier work of Denef, Van den Dries, Lipshitz, and Lenstra.

math.NT↗
arXiv2001

Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a rational solution to f_1=...=f_m=0. We show that, for almost all inputs, problem (I) can be done within coNP. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done via within the complexity class PP^{NP^NP}, i.e., within the third level of the polynomial hierarchy. The decidability of problem (II), even in the case m=n=2, remains open in general. Along the way, we prove results relating polynomial system solving over C, Q, and Z/pZ. We also prove a result on Galois groups associated to sparse polynomial systems which may be of independent interest. A practical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of crypto-systems.

math.NT↗
arXiv2000

Random Sparse Polynomial Systems

Let f:=(f^1,\...,f^n) be a sparse random polynomial system. This means that each f^i has fixed support (list of possibly non-zero coefficients) and each coefficient has a Gaussian probability distribution of arbitrary variance. We express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form. When U = (C^*)^n, the classical mixed volume is recovered. The main result in this paper is a bound on the probability that the condition number of f on the region U is larger than 1/epsilon. This bound depends on the integral of the mixed volume form over U, and on a certain intrinsic invariant of U as a subset of a toric manifold. Polynomials with real coefficients are also considered, and bounds for the expected number of real roots and for the condition number are given. The connection between zeros of sparse random polynomial systems, Kahler geometry, and mechanics (momentum maps) is discussed.

math.NA↗
arXiv2000

Algebraic Geometry Over Four Rings and the Frontier to Tractability

We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set, (b) the height of the zero-dimensional part of an algebraic set over C, and (c) the number of connected components of a semi-algebraic set. We also present some results which significantly lower the complexity of deciding the emptiness of hypersurface intersections over C and Q, given the truth of the Generalized Riemann Hypothesis. Furthermore, we state some recent progress on the decidability of the prefixes \exists\forall\exists and \exists\exists\forall\exists, quantified over the positive integers. As an application, we conclude with a result connecting Hilbert's Tenth Problem in three variables and height bounds for integral points on algebraic curves. This paper is based on three lectures presented at the conference corresponding to this proceedings volume. The titles of the lectures were ``Some Speed-Ups in Computational Algebraic Geometry,'' ``Diophantine Problems Nearly in the Polynomial Hierarchy,'' and ``Curves, Surfaces, and the Frontier to Undecidability.''

math.AG↗
arXiv2000

Computing Complex Dimension Faster and Deterministically

We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper quantitative estimates on rational univariate representations of roots of polynomial systems. As a corollary of the latter bounds, we considerably improve a recent algorithm of Koiran for deciding the emptiness of a hypersurface intersection over the complex numbers, given the truth of the Generalized Riemann Hypothesis (GRH).

math.AG↗