# Joseph RojasTexas A&M University | TAMU · Department of Mathematics

Joseph Rojas

Doctor of Philosophy

## About

112

Publications

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Introduction

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

**Skills and Expertise**

## Publications

Publications (112)

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...

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...

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)log3d (resp. log2+o(1)(dH)) in the classical Turing model: Our underlying algorithm correctly counts the number of ro...

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...

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...

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...

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.

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...

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...

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...

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) =...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 ...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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.

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...

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...

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...

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 ,......

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...

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...

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

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 ⩽0 in C <sup>n</sup>, find all solutions i...

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:

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...

. 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...