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J. Maurice Rojas

Publications and source records attributed to J. Maurice Rojas.

At least 37 records · Page 2Linked to original sources
arXiv2017

Probabilistic Condition Number Estimates For Real Polynomial Systems I: A Broader Family Of Distributions

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 generalize further by allowing over-determined systems. Along the way, we derive new Lipshitz estimates for polynomial maps from R^n to R^m, extending earlier work of Kellog on the case m=1, which may be of independent interest. In Part II, we study smoothed complexity and how sparsity (in the sense of restricting which monomial terms can appear) can help further improve earlier condition number estimates.

math.PR↗
arXiv2017

Metric Estimates and Membership Complexity for Archimedean Amoebae and Tropical Hypersurfaces

Given any complex Laurent polynomial $f$, $\mathrm{Amoeba}(f)$ is the image of its complex zero set under the coordinate-wise log absolute value map. We give an efficiently constructible polyhedral approximation, $\mathrm{ArchtTrop}(f)$, of $\mathrm{Amoeba}(f)$, and derive explicit upper and lower bounds, solely as a function of the number of monomial terms of $f$, for the Hausdorff distance between these two sets. We also show that deciding whether a given point lies in $\mathrm{ArchTrop}(f)$ is doable in polynomial-time, for any fixed dimension, unlike the corresponding problem for $\mathrm{Amoeba}(f)$, which is $\mathbf{NP}$-hard already in one variable. $\mathrm{ArchTrop}(f)$ can thus serve as a canonical low order approximation to start any higher order iterative polynomial system solving algorithm, such as homotopy continuation. $\mathrm{ArchTrop}(f)$ also provides an Archimedean analogue of Kapranov's Non-Archimedean Amoeba Theorem and a higher-dimensional extension of earlier estimates of Mikhalkin and Ostrowski.

math.AG↗
arXiv2016

Sparse Univariate Polynomials with Many Roots Over Finite Fields

Suppose $q$ is a prime power and $f\in\mathbb{F}_q[x]$ is a univariate polynomial with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of $2(q-1)^{\frac{t-2}{t-1}}$ on the number of cosets in $\mathbb{F}^*_q$ needed to cover the roots of $f$ in $\mathbb{F}^*_q$. Here, we give explicit $f$ with root structure approaching this bound: For $q$ a $(t-1)$-st power of a prime we give an explicit $t$-nomial vanishing on $q^{\frac{t-2}{t-1}}$ distinct cosets of $\mathbb{F}^*_q$. Over prime fields $\mathbb{F}_p$, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots. Nevertheless, assuming the Generalized Riemann Hypothesis, we find explicit trinomials having $Ω\left(\frac{\log p}{\log \log p}\right)$ distinct roots in $\mathbb{F}_p$.

math.NT↗
arXiv2013

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

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 time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.

math.AG↗
arXiv2013

Counting Tropically Degenerate Valuations and p-adic Approaches to the Hardness of the Permanent

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 discuss new upper bounds on the number of $p$-adic valuations of roots of certain sparse polynomial systems, culminating in a connection between quantitative p-adic geometry and complexity theory.

math.NT↗
arXiv2012

Fewnomial Systems with Many Roots, and an Adelic Tau Conjecture

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 with number of roots approaching the best known upper bounds. We also briefly review the background behind such bounds, and their application, including connections to computational number theory and variants of the Shub-Smale tau-Conjecture and the P vs. NP Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.

math.AG↗
arXiv2012

Sub-Linear Root Detection, and New Hardness Results, for Sparse Polynomials Over Finite Fields

We present a deterministic 2^O(t)q^{(t-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 F_q. A corollary of our method --- the first with complexity sub-linear in q when t is fixed --- is that the nonzero roots in F_q can be partitioned into at most 2 \sqrt{t-1} (q-1)^{(t-2)(t-1)} cosets of two subgroups S_1,S_2 of F^*_q, with S_1 in S_2. Another corollary is the first deterministic sub-linear algorithm for detecting common degree one factors of k-tuples of t-nomials in F_q[x] when k and t are fixed. When t is not fixed we show that each of the following problems is NP-hard with respect to BPP-reductions, even when p is prime: (1) detecting roots in F_p for f, (2) deciding whether the square of a degree one polynomial in F_p[x] divides f, (3) deciding whether the discriminant of f vanishes, (4) deciding whether the gcd of two t-nomials in F_p[x] has positive degree. Finally, we prove that if the complexity of root detection is sub-linear (in a refined sense), relative to the straight-line program encoding, then NEXP is not in P/Poly.

math.NT↗
arXiv2011

Randomization, Sums of Squares, and Faster Real Root Counting for Tetranomials and Beyond

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 squares and A-discriminants, including explicit obstructions to expressing positive definite sparse polynomials as sums of squares of few sparse polynomials. Our key tool is the introduction of efficiently computable chamber cones, bounding regions in coefficient space where the number of real roots of f can be computed easily. Much of our theory extends to n-variate (n+3)-nomials.

math.AG↗
arXiv2010

Optimization and NP_R-Completeness of Certain Fewnomials

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 case of polynomials (i.e., integer exponents), the log of our condition number is quadratic in the sparse encoding. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. Along the way, we extend the theory of A-discriminants to real exponents and certain exponential sums, and find new and natural NP_R-complete problems.

math.AG↗
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

Refined Asymptotics for Multigraded Sums of Squares

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 nonnegative polynomials are in fact sums of squares of polynomials remains an open problem. Blekherman was recently able to show that for degree k polynomials in n variables -- with k>=4 fixed -- those that are SOS occupy a vanishingly small fraction of those that are nonnegative on R^n, as n tends to infinity. With an eye toward the case of small n, we refine Blekherman's bounds by incorporating the underlying Newton polytope, simultaneously sharpening some of his older bounds along the way. Our refined asymptotics show that certain Newton polytopes may lead to families of polynomials where efficient SDP can still be used for most inputs.

math.AG↗
arXiv2010

New Multiplier Sequences via Discriminant Amoebae

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 by Gelfand, Kapranov, and Zelevinsky, we introduce two new natural classes of polynomials and describe diagonal operators preserving these new classes. A pleasant circumstance in our description is that these classes have a simple explicit description, one of them coinciding with the class of log-concave sequences.

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↗
arXiv2009

Faster Real Feasibility via Circuit Discriminants

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 complexity of detecting real roots for n-variate (n+k(n))-nomials transitions from P to NP-hardness as n tends to infinity. Our proofs follow in large part from a new complexity threshold for deciding the vanishing of A-discriminants of n-variate (n+k(n))-nomials. Diophantine approximation, through linear forms in logarithms, also arises as a key tool.

math.AG↗
arXiv2007

New Complexity Bounds for Certain Real Fewnomial Zero Sets

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 O(n^{11}) upper bound for the number of diffeotopy types of the real zero set of an n-variate polynomial with n+4 monomial terms.

math.AG↗
arXiv2007

Efficiently Detecting Torsion Points and Subtori

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 still hold even under certain failures of the Generalized Riemann Hypothesis, such as the presence of Siegel-Landau zeroes. Furthermore, we give a similar (but UNconditional) complexity upper bound for n=1. Finally, letting T be any algebraic subgroup of (C^*)^n we show that deciding whether X contains T is coNP-complete (relative to an even more efficient encoding),unconditionally. We thus obtain new non-trivial families of multivariate polynomial systems where deciding the existence of complex roots can be done unconditionally in the polynomial hierarchy -- a family of complexity classes lying between PSPACE and P, intimately connected with the P=?NP Problem. We also discuss a connection to Laurent's solution of Chabauty's Conjecture from arithmetic geometry.

math.AG↗
arXiv2007

On the Sharpness of fewnomial bound and the number of components of a fewnomial hypersurface

We show the existence of systems of n polynomial equations in n variables, with a total of n+k+1 distinct monomial terms, possessing [n/k+1]^k nondegenerate positive solutions. (Here, [x] is the integer part of a positive number x.) This shows that the recent upper bound of (e^2+3)/4 2^{\binom{k}{2}} n^k for the number of nondegenerate positive solutions is asymptotically sharp for fixed k and large n. We also adapt a method of Perrucci to show that there are fewer than (e^2+3)/4 2^{\binom{k}{2}} 2^n n^k connected components in a smooth hypersurface in the positive orthant of R^n defined by a polynomial with n+k+1 monomials. Our results hold for polynomials with real exponents.

math.AG↗