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13
Algebraic Geometry Over Four Rings and the Frontier to Tractability
 CONTEMPORARY MATHEMATICS
"... 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 zerodimensional part of an algeb ..."
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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 zerodimensional part of an algebraic set over C (c) the number of connected components of a semialgebraic 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 989 and 9989, 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
Some SpeedUps and Speed Limits for Real Algebraic Geometry
 Journal of Complexity, FoCM 1999 special issue
, 2000
"... this paper. The Adiscriminant in fact contains all known multivariate resultants and discriminants as special cases, and also appears in residue theory and hypergeometric functions [GKZ94]. Thus, a corollary of our last main result is that sparse elimination theory, even in low dimensions, might l ..."
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this paper. The Adiscriminant in fact contains all known multivariate resultants and discriminants as special cases, and also appears in residue theory and hypergeometric functions [GKZ94]. Thus, a corollary of our last main result is that sparse elimination theory, even in low dimensions, might lie beyond the reach of P
The Number of Real Roots of a Bivariate Polynomial on a Line,” submitted for publication
"... We prove that a polynomial f ∈ R[x, y] with t nonzero terms, restricted on a real line y = ax+b, either has at most 6t−4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether a linear polynomial y−ax−b ∈ K[x, y] divides a sparse polynomial f ..."
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Cited by 5 (0 self)
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We prove that a polynomial f ∈ R[x, y] with t nonzero terms, restricted on a real line y = ax+b, either has at most 6t−4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether a linear polynomial y−ax−b ∈ K[x, y] divides a sparse polynomial f ∈ K[x, y] with t terms in [log(H(f)H(a)H(b))[K: Q] log(deg(f))t] O(1) bit operations, where K is a real number field. 1
Detecting lacunary perfect powers and computing their roots
, 2009
"... We consider the problem of determining whether a lacunary (also called a sparse or supersparse) polynomial f is a perfect power, that is, f = h r for some other polynomial h and r ∈ N, and of finding h and r should they exist. We show how to determine if f is a perfect power in time polynomial in t ..."
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We consider the problem of determining whether a lacunary (also called a sparse or supersparse) polynomial f is a perfect power, that is, f = h r for some other polynomial h and r ∈ N, and of finding h and r should they exist. We show how to determine if f is a perfect power in time polynomial in the number of nonzero terms of f, and in terms of log deg f, i.e., polynomial in the size of the lacunary representation. The algorithm works over Fq[x] (for large characteristic) and over Z[x], where the cost is also polynomial in log ‖f‖∞. We also give a Monte Carlo algorithm to find h if it exists, for which our proposed algorithm requires polynomial time in the output size, i.e., the sparsity and height of h. Conjectures of Erdös and Schinzel, and recent work of Zannier, suggest that h must be sparse. Subject to a slightly stronger conjectures we give an extremely efficient algorithm to find h via a form of sparse Newton iteration. We demonstrate the efficiency of these algorithms with an implementation using the C++ library NTL. 1.
Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy
 J. COMPUT. SYSTEM SCI., STOC ’99 SPECIAL ISSUE
, 1999
"... We consider the averagecase complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I Given a polynomial f 2Z[v;x; y], decide the sentence 9v 8x 9y f(v; x; y) ..."
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We consider the averagecase complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I Given a polynomial f 2Z[v;x; y], decide the sentence 9v 8x 9y f(v; x; y)
Finiteness for Arithmetic Fewnomial Systems
, 2001
"... Suppose L is any finite algebraic extension of either the ordinary rational numbers or the padic 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 ..."
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Suppose L is any finite algebraic extension of either the ordinary rational numbers or the padic 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.
Supersparse black box rational function interpolation
 Manuscript
, 2011
"... We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of nonzero terms and an upper bound on the ..."
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We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of nonzero terms and an upper bound on the degree. The result is found by interpolating the rational function modulo a small prime p, and then applying an effective version of Dirichlet’s Theorem on primes in an arithmetic progression progressively lift the result to larger primes. Eventually we reach a prime number that is larger than the inputted degree bound and we can recover the original function exactly. In a variant, the initial prime p is large, but the exponents of the terms are known modulo larger and larger factors of p − 1. The algorithm, as presented, is conjectured to be polylogarithmic in the degree, but exponential in the number of terms. Therefore, it is very effective for rational functions with a small number of nonzero terms, such as the ratio of binomials, but it quickly becomes ineffective for a high number of terms. The algorithm is oblivious to whether the numerator and denominator have a common factor. The algorithm will recover the sparse form of the rational function, rather than the reduced form, which could be dense. We have experimentally tested the algorithm in the case of under 10 terms in numerator and denominator combined and observed its conjectured high efficiency.
On the Complexity of Diophantine Geometry in Low Dimensions
 Proceedings of the 31 st Annual ACM Symposium on Theory of Computing (STOC '99
"... Abstract. We consider the averagecase complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved in PSPACE: I. Given polynomials f1,..., fm ∈ Z[x1,..., xn] defining a variety of dimension ≤ 0 in C n, find all solution ..."
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Abstract. We consider the averagecase complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved in PSPACE: I. Given polynomials f1,..., fm ∈ Z[x1,..., xn] defining a variety of dimension ≤ 0 in C n, find all solutions in Z n of f1 = · · · =fm = 0. II. For a given polynomial f ∈ Z[v, x, y] defining an irreducible nonsingular nonruled surface in C 3, decide the sentence ∃v ∀x ∃y f(v, x, y) ? =0, quantified over N. Better still, we show that the truth of the Generalized Riemann Hypothesis (GRH) implies that detecting roots in Q n for the polynomial systems in problem (I) can be done via a tworound ArthurMerlin protocol, i.e., well within the second level of the polynomial hierarchy. (Problem (I) is, of course, undecidable without the dimension assumption.) The decidability of problem (II) was previously unknown. Along the way, we also prove new complexity and size bounds for solving polynomial systems over C and Z/pZ. A practical point of interest is that the aforementioned Diophantine problems should perhaps be avoided in the construction of cryptosystems.
SubLinear Root Detection, and New Hardness Results, for Sparse Polynomials Over Finite Fields, submitted for publication, also available as Math ArXiV preprint arXiv:1204.1113
"... We present a deterministic 2 O(t) q t−2 t−1 +o(1) algorithm to decidewhetheraunivariatepolynomialf, withexactlytmonomial terms and degree
Abstract  Cited by 1 (1 self)  Add to MetaCartWe present a deterministic 2 O(t) q t−2 t−1 +o(1) algorithm to decidewhetheraunivariatepolynomialf, withexactlytmonomial terms and degree <q, has a root in Fq. 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 Fq of any tnomial: the nonzero roots always admit a partition into no more than 2 √ t−1(q−1) t−2 t−1 cosets of two subgroups S1 ⊆ S2 of F ∗ q. This can be thought of as a finite field analogue of Descartes ’ Rule. A corollary of our results is the first deterministic sublinear algorithm for detecting common degree one factors of ktuples of tnomials in Fq[x] when k and t are fixed. When t is not fixed we show that, for p prime, detecting roots in Fp for f is NPhard with respect to BPPreductions. Finally, we prove that if the complexity of root detection is sublinear (in a refined sense), relative to the straightline program encoding, then NEXP⊆P/poly.
A Hitting Set Construction, with Applications to Arithmetic Circuit Lower Bounds
, 2009
"... Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential l ..."
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Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential lower bound for representations of polynomials such as n 2 i=1 (Xi − 1) under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the “hardness from derandomization” approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers. 1