Sharp estimates for triangular sets

We study the triangular representation of zero-dimensional varieties defined over the rational field (resp. a rational function field). We prove polynomial bounds in terms of intrinsic quantities for the height (resp. degree) of the coefficients of such triangular sets, whereas previous bounds were exponential. We also introduce a rational form of triangular representation, for which our estimates become linear. Experiments show the practical interest of this new representation.

[1]  Patrice Philippon,et al.  Sur des hauteurs alternatives. I , 1991 .

[2]  Joos Heintz,et al.  Corrigendum: Definability and Fast Quantifier Elimination in Algebraically Closed Fields , 1983, Theor. Comput. Sci..

[3]  Éric Schost,et al.  Complexity results for triangular sets , 2003, J. Symb. Comput..

[4]  Marc Giusti,et al.  A Gröbner Free Alternative for Polynomial System Solving , 2001, J. Complex..

[5]  Dongming Wang,et al.  Elimination Methods , 2001, Texts and Monographs in Symbolic Computation.

[6]  Éric Schost,et al.  Computing Parametric Geometric Resolutions , 2003, Applicable Algebra in Engineering, Communication and Computing.

[7]  Marc Moreno Maza,et al.  On the Theories of Triangular Sets , 1999, J. Symb. Comput..

[8]  Daniel Lazard,et al.  Solving Zero-Dimensional Algebraic Systems , 1992, J. Symb. Comput..

[9]  Marc Giusti,et al.  Lower bounds for diophantine approximations , 1997 .

[10]  Annick Valibouze,et al.  Using Galois Ideals for Computing Relative Resolvents , 1998, J. Symb. Comput..

[11]  P. Philippon,et al.  Sur des hauteurs alternatives III , 1995 .

[12]  Eric Schost.  Sur la resolution des systemes polynomiaux a parametres , 2000 .

[13]  Marie-Françoise Roy,et al.  Zeros, multiplicities, and idempotents for zero-dimensional systems , 1996 .

[14]  Joos Heintz,et al.  Estimaciones para el Teorema de Ceros de Hilbert , 1998 .

[15]  Giovanni Gallo,et al.  Efficient algorithms and bounds for Wu-Ritt characteristic sets , 1991 .

[16]  Fabrice Rouillier,et al.  Solving Zero-Dimensional Systems Through the Rational Univariate Representation , 1999, Applicable Algebra in Engineering, Communication and Computing.

[17]  Agnes Szanto,et al.  Computation with polynomial systems , 1999 .

[18]  Jean-Benoît Bost,et al.  Heights of projective varieties and positive Green forms , 1994 .

[19]  Teresa Krick,et al.  Sharp estimates for the arithmetic Nullstellensatz , 1999, math/9911094.

[20]  Evelyne Hubert,et al.  Notes on Triangular Sets and Triangulation-Decomposition Algorithms I: Polynomial Systems , 2001, SNSC.

[21]  J. E. Morais,et al.  On the intrinsic complexity of the arithmetic Nullstellensatz , 2000 .

[22]  Éric Schost,et al.  Construction of Secure Random Curves of Genus 2 over Prime Fields , 2004, EUROCRYPT.