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29
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 109 (19 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
On triangular decompositions of algebraic varieties
 Presented at the MEGA2000 Conference
, 1999
"... We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifti ..."
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Cited by 81 (35 self)
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We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifting property for calculations modulo regular chains. This allows a good management of the intermediate computations, as confirmed by several implementations and applications of this work. Our algorithm is also well suited for parallel execution.
LOWER BOUNDS FOR DIOPHANTINE APPROXIMATIONS
, 1996
"... We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. ¿From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation ..."
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Cited by 70 (27 self)
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We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. ¿From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero–dimensional polynomial equation system. This result represents a multivariate version of Liouville’s classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight–line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton’s algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.
When polynomial equation systems can be "solved" fast?
 IN PROC. 11TH INTERNATIONAL SYMPOSIUM APPLIED ALGEBRA, ALGEBRAIC ALGORITHMS AND ERRORCORRECTING CODES, AAECC11
, 1995
"... We present a new method for solving symbolically zerodimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straightline programs with FOR gates. For sequential time complexity measu ..."
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Cited by 66 (20 self)
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We present a new method for solving symbolically zerodimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straightline programs with FOR gates. For sequential time complexity measured by the size of these networks we obtain the following result: it is possible to solve any affine or toric zerodimensional equation system in nonuniform sequential time which is polynomial in the length of the input description and the &quot;geometric degree &quot; of the equation system. Here, the input is thought to be given by a straightline program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). Geometric degree has to be adequately defined. It is always bounded by the algebraiccombinatoric &quot;B'ezout number &quot; of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric degree is much smaller than
Straightline programs in geometric elimination theory
 J. Pure Appl. Algebra
, 1998
"... Dedicated to Volker Strassen for his work on complexity We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight–line prog ..."
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Cited by 65 (17 self)
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Dedicated to Volker Strassen for his work on complexity We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight–line programs. For sequential time complexity measured by network size we obtain the following result: it is possible to solve any affine or toric zero–dimensional equation system in non–uniform sequential time which is polynomial in the length of the input description and the “geometric degree ” of the equation system. Here, the input is thought to be given by a straight–line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). The geometric degree of the input system has to be adequately defined. It is always bounded by the algebraic–combinatoric “Bézout number ” of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric
Polar varieties and efficient real elimination
 MATH. Z
, 2001
"... Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations th ..."
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Cited by 42 (16 self)
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Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S0. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1,..., fp and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f1,..., fp.
Sharp estimates for the arithmetic Nullstellensatz
 Duke Math. J
"... We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which ..."
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Cited by 34 (4 self)
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We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine
Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case
 PROCEEDINGS OF THE 3RD CONFERENCE APPROXIMATION AND OPTIMIZATION IN THE CARIBBEAN, IN: APORTACIONES MATEMÁTICAS, MEXICAN SOCIETY OF MATHEMATICS
, 1998
"... The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo [10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bou ..."
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Cited by 33 (5 self)
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The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo [10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in [10] yields a method for symbolically solving a zerodimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straightline programs. The algorithm solves any affine zerodimensional equation system in nonuniform sequential time that is polynomial in the length of the input description and an adequately defined affine degree of the equation system. Replacing the affine degree of the equation system by a suitably defined real degree of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straightline program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.
Polar varieties, real equation solving and data structures: The hypersurface case
 J. COMPLEXITY
, 1997
"... In this paper we apply for the rst time a new method for multivariate equation solving which was developed in [18], [19], [20] for complex root determination to the real case. Our main result concerns the problem of nding at least one representative point foreachconnected component of a real compac ..."
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Cited by 29 (15 self)
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In this paper we apply for the rst time a new method for multivariate equation solving which was developed in [18], [19], [20] for complex root determination to the real case. Our main result concerns the problem of nding at least one representative point foreachconnected component of a real compact and smooth hypersurface. The basic algorithm of [18], [19], [20] yields a new method for symbolically solving zerodimensional polynomial equation systems over the complex numbers. feature of central importance of this algorithm is the use of a problem{adapted data type represented by the data structures arithmetic network and straightline program (arithmetic circuit). The algorithm nds the complex solutions of any a ne zerodimensional equation system in nonuniform sequential time that is polynomial in the length of the input (given in straight{line program representation) and an adequately de ned geometric degree of the equation system. Replacing the notion of geometric degree of the given polynomial equation system by a suitably de ned real (or complex) degree of certain polar varieties associated to
On the TimeSpace Complexity of Geometric Elimination Procedures
, 1999
"... In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new ge ..."
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Cited by 29 (19 self)
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In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new geometric invariant, called the degree of the input system, and the proof that the most common elimination problems have time complexity which is polynomial in this degree and the length of the input.