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Asymptotic Behaviour of the Degree of Regularity of SemiRegular Polynomial Systems
 In MEGA’05, 2005. Eighth International Symposium on Effective Methods in Algebraic Geometry
"... We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] ..."
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Cited by 45 (25 self)
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We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] family of algorithms used by the cryptographic community. 1 Motivations and Results The worstcase complexity of Gröbner bases has been the object of extensive studies. In the most general case, it is well known after work by Mayr and Meyer that the complexity is doubly exponential in the number of variables. For subclasses of polynomial systems, the complexity may be much smaller. Of particular importance is the class of regular sequences of polynomials. There, it is known that after a generic linear change of variables the complexity of the computation for the degreereverselexicographic order is simply exponential in the number of variables. Moreover, in characteristic 0, these systems are generic. Our goal is to give similar complexity bounds for overdetermined systems, for a class of systems that we
Coefficients for the FarrellJones conjecture
 Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 402
"... Abstract. We introduce the FarrellJones Conjecture with coefficients in an additive category with Gaction. This is a variant of the FarrellJones Conjecture about the algebraic K or LTheory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with ..."
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Cited by 30 (11 self)
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Abstract. We introduce the FarrellJones Conjecture with coefficients in an additive category with Gaction. This is a variant of the FarrellJones Conjecture about the algebraic K or LTheory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the setup with coefficients we obtain new results about the original FarrellJones Conjecture. The conjecture with coefficients implies the fibered version of the FarrellJones Conjecture. 1.
Some differentials on KhovanovRozansky homology
"... Abstract. We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N> 0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KRhomology of kno ..."
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Abstract. We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N> 0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KRhomology of knots with 9 crossings or fewer. 1.
Covers of the multiplicative group of an algebraically closed field of characteristic
"... Consider the classical universal cover of the one dimensional complex torus C ∗ , which gives us the exact sequence 0 − → Z i ..."
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Consider the classical universal cover of the one dimensional complex torus C ∗ , which gives us the exact sequence 0 − → Z i
Fast algorithms for zerodimensional polynomial systems using duality
 APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
, 2001
"... Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the q ..."
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Cited by 16 (3 self)
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Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the Amodule structure of the dual space � A. An important feature of our algorithms is that we do not require � A to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 n D 5/2) and O(n2 n D 5/2) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.
The density of prime divisors in the arithmetic dynamics of quadratic polynomials
 J. Lond. Math. Soc
"... Abstract. Let f ∈ Z[x], and consider the recurrence given by an = f(an−1), with a0 ∈ Z. Denote by P(f, a0) the set of prime divisors of this recurrence, i.e., the set of primes p dividing some nonzero an, and denote the natural density of this set by D(P(f, a0)). The problem of determining D(P(f, a ..."
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Cited by 16 (4 self)
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Abstract. Let f ∈ Z[x], and consider the recurrence given by an = f(an−1), with a0 ∈ Z. Denote by P(f, a0) the set of prime divisors of this recurrence, i.e., the set of primes p dividing some nonzero an, and denote the natural density of this set by D(P(f, a0)). The problem of determining D(P(f, a0)) when f is linear has attracted significant study, although it remains unresolved in full generality. In this paper we consider the case of f quadratic, where previously D(P(f, a0)) was known only in a few cases. We show D(P(f, a0)) = 0 regardless of a0 for four infinite families of f, including f = x 2 + k, k ∈ Z\{−1}. The proof relies on tools from group theory and probability theory to formulate a sufficient condition for D(P(f, a0)) = 0 in terms of arithmetic properties of the forward orbit of the critical point of f. This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critical point have been shown to determine many global dynamical properties of a quadratic polynomial. The article also includes apparently new work on the irreducibility of iterates of quadratic polynomials. 1.
A computational approach to pocklington certificates in type theory
 In Proc. of the 8th Int. Symp. on Functional and Logic Programming, volume 3945 of LNCS
, 2006
"... Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on ..."
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Cited by 15 (5 self)
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Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on computations inside and outside of the system (twolevel approach). 1 Formal Computational Proofs 1.1 Machines and the Quest for Correctness It is generally considered that modern mathematical logic was born towards the end of 19 th century, with the work of logicians like Frege, Peano, Russell or Zermelo, which lead to the precise definition of the notion of logical deduction and to formalisms like arithmetic, set theory or early type theory. From then on, a mathematical proof could be understood as a mathematical object itself, whose correction obeys some welldefined syntactical rules. In most formalisms, a formal proof is viewed as some treestructure; in natural deduction for instance, given to formal proofs σA and σB respectively of propositions A and B, these can be combined in order to build a proof of A ∧ B: σA σB ⊢ A ⊢ B ⊢ A ∧ B To sum things up, the logical point of view is that a mathematical statement holds in a given formalism if there exists a formal proof of this statement which follows the syntactical rules of the formalism. A traditional mathematical text can then be understood as an informal description of the formal proof. Things changed in the 1960ties, when N.G. de Bruijn’s team started to use computers to actually build formal proofs and verify their correctness. Using the fact that datastructures like formal proofs are very naturally represented in a computer’s memory, they delegated the proofverification work to the machine; their software Automath is considered as the first proofsystem and is the common
Resultantbased method for plane curves intersection problems
 In Proceedings of the Conference on Computer Algebra in Scientific Computing, Volume 3718 of LNCS
, 2005
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A finiteness theorem for canonical heights attached to rational maps over function fields
 J. REINE ANGEW. MATH
, 2007
"... Let K be a function field, let ϕ ∈ K(T) be a rational map of degree d ≥ 2 defined over K, and suppose that ϕ is not isotrivial. In this paper, we show that a point P ∈ P 1 ( ¯ K) has ϕcanonical height zero if and only if P is preperiodic for ϕ. This answers affirmatively a question of Szpiro and T ..."
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Let K be a function field, let ϕ ∈ K(T) be a rational map of degree d ≥ 2 defined over K, and suppose that ϕ is not isotrivial. In this paper, we show that a point P ∈ P 1 ( ¯ K) has ϕcanonical height zero if and only if P is preperiodic for ϕ. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists ε> 0 such that the set of points P ∈ P 1 (K) with ϕcanonical height at most ε is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green’s functions gϕ,v(x, y) attached to ϕ at each place v of K. For example, we show that every conjugate of ϕ has bad reduction at v if and only if gϕ,v(x, x)> 0 for all x ∈ P 1 Berk,v, where P1 Berk,v denotes the Berkovich projective line over the completion of ¯ Kv. In an appendix, we use a similar method to give a new proof of the MordellWeil theorem for elliptic curves over K.