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18
Quantum algorithms for algebraic problems
, 2008
"... Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational pro ..."
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Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum
Computing zeta functions over finite fields
 Contemporary Math
, 1999
"... Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subj ..."
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Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subject Classification: 11Y16, 11T99, 14Q15. 1.
Factoring bivariate lacunary polynomials without heights
, 2013
"... We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap theorem which allows to test whether P (X) =∑k j=1 ajX αj (1+X)βj is identically zero in polynomial time. The algorithm we obtain is more elementary than the one by Kaltofen and ..."
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We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap theorem which allows to test whether P (X) =∑k j=1 ajX αj (1+X)βj is identically zero in polynomial time. The algorithm we obtain is more elementary than the one by Kaltofen and Koiran (ISSAC’05) since it relies on the valuation of polynomials of the previous form instead of the height of the coefficients. As a result, it can be used to find some linear factors of bivariate lacunary polynomials over a field of large finite characteristic in probabilistic polynomial time.
SubLinear Root Detection, and New Hardness Results, for Sparse Polynomials Over Finite Fields
, 2013
"... 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 Fq. Our method is the first with complexity sublinear in q when t is fixed. We also prove a structural property for the nonzero root ..."
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Cited by 6 (2 self)
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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 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.
Algorithms for modular counting of roots of multivariate polynomials
 Algorithmica
, 2008
"... Given a multivariate polynomial P (X1,..., Xn) over a finite field Fq, letN(P) denote the number of roots over Fnq. The modular root counting problem is given a modulus r, to determine Nr(P) = N(P) mod r. We study the complexity of computing Nr(P), when the polynomial is given as a sum of monomial ..."
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Given a multivariate polynomial P (X1,..., Xn) over a finite field Fq, letN(P) denote the number of roots over Fnq. The modular root counting problem is given a modulus r, to determine Nr(P) = N(P) mod r. We study the complexity of computing Nr(P), when the polynomial is given as a sum of monomials. We give an efficient algorithm to computeNr(P)when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing Nr(P) is NPhard. We present some hardness results which imply that that our algorithm is essentially optimal for prime fields. We show an equivalence between maximumlikelihood decoding for ReedSolomon codes and a rootfinding problem for symmetric polynomials. 1
Faster padic Feasibility for Certain Multivariate Sparse Polynomials
"... Wepresentalgorithmsrevealingnewfamiliesofpolynomialsadmittingsubexponentialdetection of padic rational roots, relative to the sparse encoding. For instance, we prove NPcompleteness for the case of honest nvariate (n+1)nomials and, for certain special cases with p exceeding the Newton polytope v ..."
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Cited by 3 (2 self)
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Wepresentalgorithmsrevealingnewfamiliesofpolynomialsadmittingsubexponentialdetection of padic rational roots, relative to the sparse encoding. For instance, we prove NPcompleteness for the case of honest nvariate (n+1)nomials and, for certain special cases with p exceeding the Newton polytope volume, constanttime complexity. Furthermore, using the theory of linear forms in padic logarithms, we prove that the case of trinomials in one variable can be done in NP. The best previous complexity upper bounds for all these problems were EXPTIME or worse.Finally,weprovethatdetectingpadicrationalrootsforsparsepolynomialsinonevariable is NPhard 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 padic rational roots for nvariate sparse polynomials is NPhard appears to have been unknown.
Modular Counting of Rational Points over Finite Fields
"... Let Fq be the finite field of q elements, where q = p h. Let f(x) be a polynomial over Fq in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in Fq. In this paper, we give a deterministic algorithm which computes the reduction of N(f) modulo p ..."
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Let Fq be the finite field of q elements, where q = p h. Let f(x) be a polynomial over Fq in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in Fq. In this paper, we give a deterministic algorithm which computes the reduction of N(f) modulo p b in O(n(8m) (h+b)p) bit operations. This is singly exponential in each of the parameters {h, b, p}, answering affirmatively an open problem proposed in [5]. 1
The multivariate resultant is NPhard in any characteristic,” pp. 477– 488
 in P. Hliněn´y and A. Kučera (Eds), Mathematical Foundations of Computer Science (MFCS), Lecture Notes in Computer Science, 6281
, 2010
"... Abstract. The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if t ..."
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Abstract. The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system is satisfiable). In this paper we present several NPhardness results for testing whether a multivariate resultant vanishes, or equivalently for deciding whether a square system of homogeneous equations is satisfiable. Our main result is that testing the resultant for zero is NPhard under deterministic reductions in any characteristic, for systems of lowdegree polynomials with coefficients in the ground field (rather than in an extension). We also observe that in characteristic zero, this problem is in the ArthurMerlin class AM if the generalized Riemann hypothesis holds true. In positive characteristic, the best upper bound remains PSPACE. 1
On the Complexity of the Multivariate Resultant
"... The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system ..."
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The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant. First, we study the complexity of testing the multivariate resultant for zero. Our main result is that this problem is NPhard under deterministic reductions in any characteristic, for systems of lowdegree polynomials with coefficients in the ground field (rather than in an extension). In null characteristic, we observe that this problem is in the ArthurMerlin class AM if the generalized Riemann hypothesis holds true, while the best known upper bound in positive characteristic remains PSPACE. Second, we study the classical algorithms to compute the resultant. They usually rely on the computation of the determinant of an exponentialsize matrix, known as Macaulay matrix. We show that this matrix belongs to a class of succinctly representable matrices, for which testing the determinant for zero is proved PSPACEcomplete. This means that improving Canny’s PSPACE upper bound requires either to look at the fine structure of the Macaulay matrix to find an ad hoc algorithm for computing its determinant, or to use altogether different techniques. ∗Corresponding author. †This work was partially funded by the European Community (7th PCRD Contract: