Results 1  10
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10
Greatest Common Divisors of Polynomials Given by StraightLine Programs
 J. ACM
, 1988
"... . F Algorithms on multivariate polynomials represented by straightline programs are developed irst it is shown that most algebraic algorithms can be probabilistically applied to data that is given by y r a straightline computation. Testing such rational numeric data for zero, for instance, is faci ..."
Abstract

Cited by 51 (18 self)
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. F Algorithms on multivariate polynomials represented by straightline programs are developed irst it is shown that most algebraic algorithms can be probabilistically applied to data that is given by y r a straightline computation. Testing such rational numeric data for zero, for instance, is facilitated b andom evaluations modulo random prime numbers. Then auxiliary algorithms are constructed that a determine the coefficients of a multivariate polynomial in a single variable. The first main result is an lgorithm that produces the greatest common divisor of the input polynomials, all in straightline r a representation. The second result shows how to find a straightline program for the reduced numerato nd denominator from one for the corresponding rational function. Both the algorithm for that conl c struction and the greatest common divisor algorithm are in random polynomialtime for the usua oefficient fields and output a straightline program, which with controllably high probab...
DAGWOOD  A System for Manipulating Polynomial Given by StraightLine Programs
 ACM Trans. Math. Software
, 1988
"... . We discuss the design, implementation, and benchmarking of a system tha an manipulate symbolic expressions represented by their straightline computations. Our syst c tem is capable of performing rational arithmetic on, evaluating, differentiating, taking greates ommon divisors of, and factoring ..."
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Cited by 19 (8 self)
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. We discuss the design, implementation, and benchmarking of a system tha an manipulate symbolic expressions represented by their straightline computations. Our syst c tem is capable of performing rational arithmetic on, evaluating, differentiating, taking greates ommon divisors of, and factoring polynomials in straightline format. The straightline c results can also be converted to standard sparse format. We show by example that our system an handle problems for which conventional methods lead to excessive intermediate expres # sion swell. ############## * This material is based upon work supported by the National Science Foundation under Grant No. DCR85s 04391 and by an IBM Faculty Development Award. Computational resources during our stay at Berkeley were upported in part by the Army Research Office under Grant No. DAAG2985K0070 through the Center for M Pure and Applied Mathematics at the University of California. This paper appears in the ACM Transactions on athematical...
On The Computational Hardness Of Testing SquareFreeness Of Sparse Polynomials
, 1999
"... We show that deciding squarefreeness of a sparse univariate polynomial over ZZ and over the algebraic closure of a finite field IFq of p elements is NPhard. We also discuss some related open problems about sparse polynomials. ..."
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Cited by 11 (1 self)
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We show that deciding squarefreeness of a sparse univariate polynomial over ZZ and over the algebraic closure of a finite field IFq of p elements is NPhard. We also discuss some related open problems about sparse polynomials.
Counting Curves and Their Projections
 Computational Complexity
, 1996
"... . Some deterministic and probabilistic methods are presented for counting and estimating the number of points on curves over finite fields, and on their projections. The classical question of estimating the size of the image of a univariate polynomial is a special case. For curves given by spars ..."
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Cited by 10 (1 self)
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. Some deterministic and probabilistic methods are presented for counting and estimating the number of points on curves over finite fields, and on their projections. The classical question of estimating the size of the image of a univariate polynomial is a special case. For curves given by sparse polynomials, the counting problem is #Pcomplete via probabilistic parsimonious Turing reductions. 1. Introduction One of the most celebrated results in algebraic geometry is Weil's theorem on the number of points on algebraic curves over a finite field. In this paper, we address some computational problems related to this question. Our main results are: ffi A "computational Weil estimate" for projections of curves and images of polynomials, in Section 3. ffi #Pcompleteness of the exact counting problem for sparse curves, in Section 4. We consider a finite field F q with q elements, an algebraic closure K of F q , a polynomial f 2 F q [x; y] of degree n , the plane curve C = ff = 0...
Short Proofs for Nondivisibility of Sparse Polynomials under the Extended Riemann Hypothesis
, 1991
"... Symbolic manipulation of sparse polynomials, given as lists of exponents and nonzero coefficients, appears to be much more complicated than dealing with polynomials in dense encoding (see e.g. [GKS 90, KT 88, P 77a, P 77b]). The first results in this direction are due to Plaisted [P 77a, P 77b], ..."
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Cited by 8 (2 self)
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Symbolic manipulation of sparse polynomials, given as lists of exponents and nonzero coefficients, appears to be much more complicated than dealing with polynomials in dense encoding (see e.g. [GKS 90, KT 88, P 77a, P 77b]). The first results in this direction are due to Plaisted [P 77a, P 77b], who proved, in particular, the NPcompleteness of divisibility of a polynomial x n \Gamma1 by a product of sparse polynomials. On the other hand, essentially nothing nontrivial is known about the complexity of the divisibility problem of two sparse integer polynomials. (One can easily prove that it is in PSPACE with the help of [M 86].) Here we prove that nondivisibility of two sparse multivariable polynomials is in NP, provided that the Extended Riemann Hypothesis (ERH) holds (see e.g. [LO 77]). The divisibility problem is closely related to the rational interpolation problem (whose decidability and complexity bound are determined in [GKS 90]). In this setting we assume that a r...
Algorithms for Sparse Rational Interpolation
 PROC. INTERN. SYMP. ON SYMBOLIC AND ALGEBRAIC COMP
, 1991
"... We present two algorithms on sparse rational interpolation. The first is the interpolation algorithm in a sense of the sparse partial fraction representation of rational functions. The second is the algorithm for computing the entier and the remainder of a rational function. The first algorithm work ..."
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Cited by 7 (5 self)
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We present two algorithms on sparse rational interpolation. The first is the interpolation algorithm in a sense of the sparse partial fraction representation of rational functions. The second is the algorithm for computing the entier and the remainder of a rational function. The first algorithm works without apriori known bound on the degree of a rational function, the second one is in the class NC provided the degree is known. The presented algorithms complement the sparse interpolation results of [Grigoriev, Karpinski, and Singer 90b].
Complexity Issues in Dynamic Geometry
 IN PROCEEDINGS OF THE SMALE FEST 2000, HONGKONG
, 2000
"... This article deals with the intrinsic complexity of tracing and reachability questions in the context of elementary geometric constructions. We consider constructions from... ..."
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Cited by 5 (5 self)
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This article deals with the intrinsic complexity of tracing and reachability questions in the context of elementary geometric constructions. We consider constructions from...
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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Cited by 3 (0 self)
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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.
Short Proofs for Nondivisibility of Sparse Polynomials under the Extended Riemann Hypothesis
, 1996
"... Symbolic manipulation of sparse polynomials, given as lists of exponents and nonzero coefficients, appears to be much more complicated than dealing with polynomials in dense encoding (see e.g. [GKS 90, KT 88, P 77a, P 77b]). The first results in this direction are due to Plaisted [P 77a, P 77b], who ..."
Abstract
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Symbolic manipulation of sparse polynomials, given as lists of exponents and nonzero coefficients, appears to be much more complicated than dealing with polynomials in dense encoding (see e.g. [GKS 90, KT 88, P 77a, P 77b]). The first results in this direction are due to Plaisted [P 77a, P 77b], who proved, in particular, the NPcompleteness of divisibility of a polynomial x n \Gamma 1 by a product of sparse polynomials. On the other hand, essentially nothing nontrivial is known about the complexity of the divisibility problem of two sparse integer polynomials. (One can easily prove that it is in PSPACE with the help of [M 86].) Here we prove that nondivisibility of two sparse multivariable polynomials is in NP, provided that the Extended Riemann Hypothesis (ERH) holds (see e.g. [LO 77]). The divisibility problem is closely related to the rational interpolation problem (whose decidability and complexity bound are determined in [GKS 90]). In this setting we assume that a rational funct...
Differential Forms in Computational Algebraic Geometry [Extended Abstract] ∗
"... We give a uniform method for the two problems #CCC and #ICC of counting connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work efficiently in parallel and can be implemented by al ..."
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We give a uniform method for the two problems #CCC and #ICC of counting connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work efficiently in parallel and can be implemented by algebraic circuits of polynomial depth, i.e., in parallel polynomial time. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Szántó [39] computing a variant of characteristic sets. The crucial complexity parameter for #ICC turns out to be the number of equations. We describe a randomised algorithm solving #ICC for a fixed number of rational equations given by straightline programs (slps), which runs in parallel polylogarithmic time in the length and the degree of the slps.