Joseph Rojas

Joseph Rojas
Texas A&M University | TAMU · Department of Mathematics

Doctor of Philosophy

About

112
Publications
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953
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Introduction
All my papers are freely downloadable from my papers page: www.math.tamu.edu/~rojas/list2.html

Publications

Publications (112)
Preprint
Full-text available
Sturm's Theorem is a fundamental 19th century result relating the number of real roots of a polynomial $f$ in an interval to the number of sign alternations in a sequence of polynomial division-like calculations. We provide a short direct proof of Sturm's Theorem, including the numerically vexing case (ignored in many published accounts) where an i...
Article
We improve the algorithms of Lauder-Wan [11] and Harvey [8] to compute the zeta function of a system of m polynomial equations in n variables, over the q element finite field Fq, for large m. The dependence on m in the original algorithms was exponential in m. Our main result is a reduction of the dependence on m from exponential to polynomial. As...
Article
Text For any fixed field K∈{Q2,Q3,Q5,…}, we prove that all univariate polynomials f with exactly 3 (resp. 2) monomial terms, degree d, and all coefficients in {±1,…,±H}, can be solved over K within deterministic time log4+o(1)⁡(dH)log3⁡d (resp. log2+o(1)⁡(dH)) in the classical Turing model: Our underlying algorithm correctly counts the number of ro...
Preprint
Full-text available
Consider a univariate polynomial f in Z[x] with degree d, exactly t monomial terms, and coefficients in {-H,...,H}. Solving f over the reals, R, in polynomial-time can be defined as counting the exact number of real roots of f and then finding (for each such root z) an approximation w of logarithmic height (log(dH))^{O(1)} such that the Newton iter...
Article
Full-text available
A correction to this paper has been published: https ://doi.org/10.1007/s00208-019-01808-5 A subsidiary result (Theorem 1.1, on the undecidability of determining membership in the amoeba of 1-x-y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs...
Preprint
Full-text available
For any fixed field $K\!\in\!\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5, \ldots\}$, we prove that all polynomials $f\!\in\!\mathbb{Z}[x]$ with exactly $3$ (resp. $2$) monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ within deterministic time $\log^{7+o(1)}(dH)$ (resp. $\log^{2+o(1)}(dH)$) in t...
Article
We consider the sensitivity of real zeros of structured polynomial systems to pertubations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems and extend these estimates to the smoothed analysis setting.
Preprint
Full-text available
Let $k,p\in \mathbb{N}$ with $p$ prime and let $f\in\mathbb{Z}[x_1,x_2]$ be a bivariate polynomial with degree $d$ and all coefficients of absolute value at most $p^k$. Suppose also that $f$ is variable separated, i.e., $f=g_1+g_2$ for $g_i\in\mathbb{Z}[x_i]$. We give the first algorithm, with complexity sub-linear in $p$, to count the number of ro...
Preprint
Full-text available
Suppose $A=\{a_1,\ldots,a_{n+2}\}\subset\mathbb{Z}^n$ has cardinality $n+2$, with all the coordinates of the $a_j$ having absolute value at most $d$, and the $a_j$ do not all lie in the same affine hyperplane. Suppose $F=(f_1,\ldots,f_n)$ is an $n\times n$ polynomial system with generic integer coefficients at most $H$ in absolute value, and $A$ th...
Article
The applications of solving systems of polynomial equations are legion: The real case permeates all of non-linear optimization as well as numerous problems in engineering. The p -adic case leads to many classical questions in number theory, and is close to many applications in cryptography, coding theory, and computational number theory. As such, i...
Article
Full-text available
We study the complexity of approximating complex zero sets of certain n-variate exponential sums. We show that the real part, R, of such a zero set can be approximated by the \((n-1)\)-dimensional skeleton, T, of a polyhedral subdivision of \(\mathbb {R}^n\). In particular, we give an explicit upper bound on the Hausdorff distance: \(\Delta (R,T) =...
Preprint
In this paper, we improve the algorithms of Lauder-Wan \cite{LW} and Harvey \cite{Ha} to compute the zeta function of a system of $m$ polynomial equations in $n$ variables over the finite field $\FF_q$ of $q$ elements, for $m$ large. The dependence on $m$ in the original algorithms was exponential in $m$. Our main result is a reduction of the expon...
Preprint
We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$, we prove that any polynomial $f\in\mathbb{Z}[x_1]$ with exactly $3$ monomial terms, degree $d$, and all coeffi...
Conference Paper
Suppose F:=(f_1,łdots,f_n) is a system of random n-variate polynomials with f_i having degree łeq\!d_i and the coefficient of x^a_1 _1\cdots x^a_n _n in f_i being an independent complex Gaussian of mean 0 and variance \fracd_i! a_1!\cdots a_n!łeft(d_i-\sum^n_j=1 a_j \right)! . Recent progress on Smale's 17þth Problem by Lairez --- building upon sem...
Preprint
Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance $\frac{d_i!}{a_1!\cdots a_n!\left(d_i-\sum^n_{j=1}a_j \right)!}$. Recent progress on Smale's 17th Problem by Laire...
Article
To whom it may concern: You can freely download all my published papers from my papers page at Texas A&M University: www.math.tamu.edu/~rojas/list2.html
Preprint
We consider the sensitivity of real zeros of polynomial systems with respect to perturbation of the coefficients, and extend our earlier probabilistic estimates for the condition number in two directions: (1) We give refined bounds for the condition number of random structured polynomial systems, depending on a variant of sparsity and an intrinsic...
Preprint
Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the number of roots of $f$ in $\mathbb{Z}/\!\left(p^k\right)$ within time $d^3(k\log p)^{2+o(1)}$. (We in fact prove a...
Article
Thanks to earlier work of Koiran, it is known that the truth of the Generalized Riemann Hypothesis (GRH) implies that the dimension of algebraic sets over the complex numbers can be determined within the polynomial-hierarchy. The truth of GRH thus provides a direct connection between a concrete algebraic geometry problem and the P vs.NP Problem, in...
Article
Full-text available
Suppose $p$ is a prime, $t$ is a positive integer, and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\!p^t$. We show that for any fixed $t$, we can compute the number of roots in $\mathbb{Z}/(p^t)$ of $f$ in deterministic time $(d+\log p)^{O(1)}$. This fixed parameter tractability appears to...
Article
Suppose $c_1,\ldots,c_{n+k}$ are real numbers, $\{a_1,\ldots,a_{n+k}\}\!\subset\!\mathbb{R}^n$ is a set of points not all lying in the same affine hyperplane, $y\!\in\!\mathbb{R}^n$, $a_j\cdot y$ denotes the standard real inner product of $a_j$ and $y$, and we set $g(y)\!:=\!\sum^{n+k}_{j=1} c_j e^{a_j\cdot y}$. We prove that, for generic $c_j$, th...
Article
We extend the notion of $\mathcal{A}$-discriminant, and Kapranov's parametrization of $\mathcal{A}$-discriminant varieties, to complex exponents. As an application, we focus on the special case where $\mathcal{A}$ is a set of $n+3$ points in $\mathbb{R}^n$ with non-defective Gale dual, $g$ is a real $n$-variate exponential sum with spectrum $\mathc...
Article
We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular - for a version of the condition number defined by Cucker, Krick, Malajovich, and Wschebor - we establish new probabilistic estimates that allow a much broader family of measures than considered earlier. We also generaliz...
Article
We present a deterministic $2^{O(t)}q^{\frac{t-2}{t-1}+o(1)}$ algorithm to decide whether a univariate polynomial $f$, with $t$ monomial terms and degree $<q$, has a root in the finite field $\mathbb{F}_q$. Our method is the first with complexity sublinear in $q$ when $t$ is fixed. We also prove a structural property for the nonzero roots in $\math...
Article
Given any multivariate exponential sum, $g$, the real part of the complex zero set of $g$ forms a sub-analytic variety $\Re(Z(g))$ generalizing the amoeba of a complex polynomial. We propose an extension, $\mathrm{Trop}(g)$, of the notion of Archimedean tropical hypersurface, in order to polyhedrally approximate $\Re(Z(g))$. We then derive explicit...
Article
Suppose $q$ is a prime power and $f$ is a univariate polynomial (over the finite field $\mathbf{F}_q$) with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas recently proved an upper bound of $2(q-1)^{(t-2)/(t-1)}$ on the number of cosets of $\mathbf{F}^*_q$ needed to cover t...
Article
Full-text available
We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the...
Article
The Shub-Smale Tau Conjecture is a hitherto unproven statement (on integer roots of polynomials) whose truth implies both a variant of $P\neq NP$ (for the BSS model over C) and the hardness of the permanent. We give alternative conjectures, some potentially easier to prove, whose truth still implies the hardness of the permanent. Along the way, we...
Article
Full-text available
Given any complex Laurent polynomial f, Amoeba(f) is the image of the complex zero set of f under the coordinate-wise log absolute value map. We give an efficiently constructible polyhedral approximation, ArchTrop(f), of Amoeba(f), and derive explicit upper and lower bounds, solely as a function of the sparsity of f, for the Hausdorff distance betw...
Article
Suppose that f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present a deterministic algorithm of complexity polynomial in log D that, for most inputs, counts the number of real roots of f . The best previous algorithms have complexity super-linear in D. We also discuss connections to sums of squares and A-discrimina...
Article
We present a deterministic 2O(t)qt-2/t-1 +o(1) algorithm to decide whether a univariate polynomial f, with exactly t monomial terms and degree q, has a root in Fq. Our method is the first with complexity sub-linear in q when tis fixed. We also prove a structural property for the nonzero roots in Fq of any t-nomial: the nonzero roots always admit a...
Article
This document presents current technical progress and dissemination of results for the Mathematics for Analysis of Petascale Data (MAPD) project titled 'Topology for Statistical Modeling of Petascale Data', funded by the Office of Science Advanced Scientific Computing Research (ASCR) Applied Math program. Many commonly used algorithms for mathemati...
Article
Full-text available
Suppose f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present an algorithm, with complexity polynomial in log D on average (relative to the stable log-uniform measure), for counting the number of real roots of f. The best previous algorithms had complexity super-linear in D. We also discuss connections to sums of s...
Article
We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n + 2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field op-erations and inequality checks, and are polynomial in n and the logarithm of a certain condition number. For the spe-ci...
Article
Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems wit...
Article
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 constant term and the Newton polytope volume, in constant time. Furthermore, using th...
Article
Full-text available
In their classic 1914 paper, Polya and Schur introduced and characterized two types of linear operators acting diagonally on the monomial basis of R[x], sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same sign) to real-rooted polynomials. Motivated by fundamental properties of amoebae and discriminants discovered b...
Article
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...
Article
Relative to the sparse encoding, we show that deciding whether a univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a sharper complexity upper bound of P for polynomials with suit-ably generic p-adic Newton polygon. We thus improve the best previous complexity upper bound of EXPTIME. We also prove an u...
Article
We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are polynomial in n and the logarithm of a certain condition number. For the special c...
Article
This is a progress report on polynomial system solving for statistical modeling. This is a progress report on polynomial system solving for statistical modeling. This quarter we have developed our first model of shock response data and an algorithm for identifying the chamber cone containing a polynomial system in n variables with n+k terms within...
Article
Full-text available
We show that detecting real roots for honestly n-variate (n+2)-nomials (with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. We then give a characterization of those functions k(n) such that the c...
Article
To prove that a polynomial is nonnegative on R^n one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnega...
Article
We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner ...
Article
We show that deciding whether a sparse polynomial in one variable has a root in F p (for p prime) is NP-hard with respect to BPP reductions. As a consequence, we answer open questions on the factorization of sparse polynomials posed by Karpinski and Shparlinski, and Cox. We also derive analogous results for detecting p-adic rational roots, thus par...
Article
Withdrawn by the authors due to an error in the proof of the finite field result (Thm. 1.5): The random primes used in the proof need NOT avoid the exceptional primes from Lemma 2.7, thus leaving Thm. 1.5 unproved. Comment: This paper has been withdrawn by the authors
Article
Consider real bivariate polynomials f and g, respectively having 3 and m monomial terms. We prove that for all m>=3, there are systems of the form (f,g) having exactly 2m-1 roots in the positive quadrant. Even examples with m=4 having 7 positive roots were unknown before this paper, so we detail an explicit example of this form. We also present an...
Chapter
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, (*) is nearly NP-comple...
Article
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-discrimina...
Article
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 rand...
Conference Paper
This paper presents a new method for coordinated motion planning of multiple mobile agents. The position in 2-D of each mobile agent is mapped to a complex number and a time varying polynomial contains information regarding the current positions of all mobile agents, the degree of the polynomial being the number of mobile agents and the roots of th...
Article
Many applications require a method for translating a large list of bond angles and bond lengths to precise atomic Cartesian coordinates. This simple but computationally consuming task occurs ubiquitously in modeling proteins, DNA, and other polymers as well as in many other fields such as robotics. To find an optimal method, algorithms can be compa...
Article
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 ε all the roots of f in the interval [0,R] using just O(log(D)log(D log R/ε)) arithmetic operations. In particular, we can count the number of roots in any bounded interval using just O(log2...
Article
Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse encoding), under a plausible assumption on primes in arithmetic progression. In particular, our hypothesis can s...
Article
Detecting degenerate object configurations is an important part of a robust boundary evaluation system. The most efficient current methods for exact boundary evaluation rely on a general position assumption, and fail in the presence of degeneracies. To address this problem, we describe here a method, based on the rational univariate reduction (RUR)...
Article
Suppose X is the complex zero set of a flnite collection of polynomials in Z(x1; :::; xn). Also let T be any multiplicative translate of an algebraic subgroup of (C ⁄ ) n . We prove that we can decide X ? ¶T (resp. X\T ? =;) within coNP (resp. NP NP ), relative to the sparse encoding. In particular, we can detect within NP NP whether X contains a p...
Article
Full-text available
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 poly...
Article
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 ε all the roots of f in the interval [0,R] using just arithmetic operations. In particular, we can count the number of roots in any bounded interval using just arithmetic operations. Our speed...
Article
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...
Article
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 no...
Article
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...
Article
Full-text available
We prove that any pair of bivariate trinomials has at most five 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...
Article
Let f≔(f1,…,fn) 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/ε. 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 underl...
Article
We derive an improved upper bound for the VC-dimension of neural networks with polynomial activation functions. This improved bound is based on a result of Rojas [Roj00] on the number of connected components of a semi-algebraic set.
Conference Paper
Consider any nonzero univariate polynomial with rational coefficients, presented as an elementary algebraic expression (using only integer exponents). Letting σ(f) denotes the additive complexity of f, we show that the number of rational roots of f is no more than 15 + σ(f)2(24.01)σ(f)σ(f) !. This provides a sharper arithmetic analogue of earlier r...
Article
In this note, we derive an improved upper bound for the VC-dimension of neural networks with polynomial activation functions. This improved bound is based on a result of Rojas on the number of connected components of a semi-algebraic set.
Article
Let L be any number field or p-adic field and consider F := (f 1 , . . . , f k ) where f i #L[x 1 1 , . . . , x 1 n ]{0} for all i and there are exactly distinct exponent vectors appearing in f 1 , . . . , f k . We prove that F has no more than 1+ #n( - n + 1) min{1,n-1} ( - n) 2 log n geometrically isolated roots in L n , where # is an explicit an...
Conference Paper
Consider any nonzero univariate polynomial with rational coefficients, presented as an elementary algebraic expression (using only integer exponents). Letting ¿(f) denotes the additive complexity of f, we show that the number of rational roots of f is no more than 15 + ¿(f)2(24.01)¿(f)¿(f)!. This provides a sharper arithmetic analogue of earlier re...
Article
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 (c) the number of connected compon...
Article
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 ,......
Article
Let f be a degree D univariate polynomial with real coecients and at most 3 monomial terms. We show that all the roots of f in any closed interval of length R can be approximated within an accuracy of " using just O(log D log log R " +log 2 D) arithmetic steps, i.e., the arithmetic complexity is polylogarithmic in the degree of the underlying compl...
Article
Full-text available
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(lo...
Article
this paper. 3 2. More generally, suppose Y is a normal variety with a smooth open subvariety U Y satisfying the following condition: locally analytically at every point, (Y; U Y ) is isomorphic to a local analytic neighborhood of some torus embedding (X; T ). We then call Y a toroidal variety and (Y; U Y ) a toroidal embedding
Article
Introduction This paper outlines a Kahler-geometric approach to the study of random sparse polynomial systems, with a view toward understanding the distribution of solutions and their numerical conditioning. Section 2 develops the necessary mathematical framework while section 3 contains our main results for random sparse polynomial systems, inclu...
Article
We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I. Given a polynomial f ∈ Z[v, x, y], decide the sentence ∃v ∀x ∃y f(v, x, y) ? = 0, with all three quantifiers ranging over N (or Z). II. Given polynomials f1,..., fm ∈Z[x1,..., xn] with m≥n, decide if there i...
Article
This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the u-resultant and then retailor the generalized characteristic polynomial to fully exploit sparsity in the monomial structure of any given polynomial system. We thus obtain a fast new algorithm for univariate reduction a...
Article
We derive an explicit formula for the expected number of real roots of certain random sparse polynomial systems. We propose (and use) a probability measure which is natural in the sense that it is invariant under a natural G-action on the space of roots, where G is a product of orthogonal groups. Our formula arose from an effort (now an ongoing pro...
Article
This brief note corrects some errors in the paper quoted in the title, highlights a combinatorial result which may have been overlooked, and points to further improvements in recent literature. 1.
Article
Let f :=(f 1 ; : : : ; f n ) be a sparse random polynomial system. This means that each f i has xed support (list of possibly non-zero coe- cients) and each coecient 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. Whe...
Article
We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I. Given a polynomial f∈[v, x, y], decide the sentence ∃v ∀x ∃yf(v, x, y)=?0, with all three quantifiers ranging over (or ). II. Given polynomials f1, …, fm∈[x1, …, xn] with m⩾n, decide if there is a rational s...
Article
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We al...
Article
Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial t...
Article
We show that the decidability of an amplification of Hilbert's Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves. We obtain this as a corollary of a new positive complexity result: the Diophantine prefix ∃∀∃ is generically decidable. This means that we give a precise geometric c...
Article
E 1 ; : : : ; En+1 be nonempty finite subsets of (N [ f0g) n . For any e = (e 1 ; : : : ; e n ) 2 (N [ f0g) n let x e denote the monomial x e1 1 Delta Delta Delta x en n . In this way we will let f 1 ; : : : ; fn+1 be polynomials in the variables fx 1 ; : : : ; xn g with (algebraically independent) indeterminate coefficients, such that the set of e...
Conference Paper
We give a new algorithm, with three versions, for computing the number of real roots of a system of n polynomial equations in n unknowns. The first version is of Monte Carlo type and, neglecting logarithmic factors, runs in time quadratic in the average number of complex roots of a closely related system. The other two versions run nearly as fast a...
Article
Full-text available
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 exte...
Article
We give new positive and negative results, some conditional, on speeding up computational algebraic geometry over the reals: 1. A new and sharper upper bound on the number of connected components of a semi-algebraic set. Our bound is novel in that it is stated in terms of the volumes of certain polytopes and, for a large class of inputs, beats the...
Article
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...
Conference Paper
We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved within PSPACE: I. Given polynomials f<sub>1</sub>,…,f<sub>m</sub>∈ Z [x<sub>1 </sub>,…,x<sub>n</sub>] defining a variety of dimension &les;0 in C <sup>n</sup>, find all solutions i...
Conference Paper
We consider the average-case complexity of some otherwise undecidable or open Dio- phantine problems. More precisely, we show that the following two problems can be solved in PSPACE:
Article
We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved in the complexity class PSPACE: (I) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] defining a variety of dimension <=0 in C^n, find all solutions in Z^n of f_1=...=f_m=0. (II) For...
Article
. We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over an algebraically closed field) of any n by n system of polynomial equations. Since we use the sparse resultan...

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