Results 1  10
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37
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 82 (16 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
Almost tight recursion tree bounds for the Descartes method
 In Proc. Int. Symp. on Symbolic and Algebraic Computation
, 2006
"... We give a unified (“basis free”) framework for the Descartes method for real root isolation of squarefree real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound ..."
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Cited by 30 (3 self)
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We give a unified (“basis free”) framework for the Descartes method for real root isolation of squarefree real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiXi with integer coefficients ai  < 2 L, this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.
Real Algebraic Numbers: Complexity Analysis and Experimentation
 RELIABLE IMPLEMENTATIONS OF REAL NUMBER ALGORITHMS: THEORY AND PRACTICE, LNCS (TO APPEAR
, 2006
"... We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both metho ..."
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Cited by 30 (17 self)
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We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non squarefree polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.
Fast Computation of Special Resultants
, 2006
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 20 (8 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
Extremal real algebraic geometry and Adiscriminants
 Moscow Mathematical Journal
, 2007
"... We present a new, far simpler family of counterexamples to Kushnirenko’s Conjecture. Along the way, we illustrate a computerassisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We ..."
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Cited by 11 (5 self)
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We present a new, far simpler family of counterexamples to Kushnirenko’s Conjecture. Along the way, we illustrate a computerassisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recent formula for the Adiscriminant, and give new bounds on the topology of certain Adiscriminant varieties. A consequence of the latter result is a new upper bound on the number of topological types of certain real algebraic sets defined by sparse polynomial equations. 1
Fast computation with two algebraic numbers
 September
, 2002
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 8 (3 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
On the Topology of Planar Algebraic Curves
"... We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e cienc ..."
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Cited by 7 (1 self)
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We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e ciency of our algorithm. 1
First Steps in Algorithmic Real Fewnomial Theory
, 2008
"... Fewnomial theory began with explicit bounds — solely in terms of the number of variables and monomial terms — on the number of real roots of systems of polynomial equations. Here we take the next logical step of investigating the corresponding existence problem: Let FEASR denote the problem of decid ..."
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Cited by 5 (5 self)
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Fewnomial theory began with explicit bounds — solely in terms of the number of variables and monomial terms — on the number of real roots of systems of polynomial equations. Here we take the next logical step of investigating the corresponding existence problem: Let FEASR denote the problem of deciding whether a given system of multivariate polynomial equations with integer coefficients has a real root or not. We describe a phasetransition for when m is large enough to make FEASR be NPhard, when restricted to inputs consisting of a single nvariate polynomial with exactly m monomial terms: polynomialtime for m≤n + 2 (for any fixed n) and NPhardness for m ≥ n + n ε (for n varying and any fixed ε> 0). Because of important connections between FEASR and Adiscriminants, we then study some new families of Adiscriminants whose signs can be decided within polynomialtime. (Adiscriminants contain all known resultants as special cases, and the latter objects are central in algorithmic algebraic geometry.) Baker’s Theorem from diophantine approximation arises as a key tool. Along the way, we also derive new quantitative bounds on the real zero
On Solving Fewnomials Over Intervals In Fewnomial Time
"... Let f be a degree D univariate polynomial with real coefficients and at most 3 monomial terms. We show that all the roots of f in any closed interval of length R can be approximated within an accuracy of # using just O(log D log log R # +log 2 D) arithmetic steps, i.e., the arithmetic complexity is ..."
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Cited by 4 (3 self)
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Let f be a degree D univariate polynomial with real coefficients and at most 3 monomial terms. We show that all the roots of f in any closed interval of length R can be approximated within an accuracy of # using just O(log D log log R # +log 2 D) arithmetic steps, i.e., the arithmetic complexity is polylogarithmic in the degree of the underlying complex variety. In particular, of independent interest is an algebraic subroutine of our algorithm which counts the roots in any given interval using just O(log 2 D) arithmetic steps. The best previous arithmetic complexity upper bounds for these respective solving and counting problems were O(D log 5 D log log R # ) and O(D log 2 D log log D).
An efficient and exact subdivision algorithm for isolating complex roots of a polynomial and its complexity analysis
, 2009
"... We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a squarefree polynomial. The subdivision predicates are based on evaluating the original polynomial or its derivatives, and hence is easy to implement. It can be seen as a generalization of a previous real root isolati ..."
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Cited by 3 (3 self)
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We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a squarefree polynomial. The subdivision predicates are based on evaluating the original polynomial or its derivatives, and hence is easy to implement. It can be seen as a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, the algorithm is applicable for general analytic functions. We provide a complexity analysis of our algorithm on the benchmark problem of isolating all complex roots of a squarefree polynomial with Gaussian integer coefficients. The analysis is based on a novel technique called δclusters. This analysis shows, somewhat surprisingly, that the simple EVAL algorithm matches (up to logarithmic factors) the bit complexity bounds of current practical exact algorithms such as those based on Descartes, Continued Fraction or Sturm methods. Furthermore, the more general CEVAL also achieves the same complexity.