TY - GEN

T1 - Size of coefficients of lexicographical Groöbner bases

T2 - 2009 International Symposium on Symbolic and Algebraic Computation, ISSAC 2009

AU - Dahan, Xavier

PY - 2009/12/1

Y1 - 2009/12/1

N2 - This work is limited to the zero-dimensional, radical, and bivariate case. A lexicographical Gröbner basis can be simply viewed as Lagrange interpolation polynomials. In the same way the Chinese remaindering theorem generalizes Lagrange interpolation, we show how a triangular decomposition is linked to a specific Gröbner basis (not the reduced one). A bound on the size of the coefficients of this specific Gröbner basis is proved using height theory, then a bound is deduced for the reduced Gröbner basis. Besides, the link revealed between the Gröbner basis and the triangular decomposition gives straightforwardly a numerical estimate to help finding a lucky prime in the context of modular methods.

AB - This work is limited to the zero-dimensional, radical, and bivariate case. A lexicographical Gröbner basis can be simply viewed as Lagrange interpolation polynomials. In the same way the Chinese remaindering theorem generalizes Lagrange interpolation, we show how a triangular decomposition is linked to a specific Gröbner basis (not the reduced one). A bound on the size of the coefficients of this specific Gröbner basis is proved using height theory, then a bound is deduced for the reduced Gröbner basis. Besides, the link revealed between the Gröbner basis and the triangular decomposition gives straightforwardly a numerical estimate to help finding a lucky prime in the context of modular methods.

KW - Gröbner bases

KW - Space complexity

KW - Triangular sets

UR - http://www.scopus.com/inward/record.url?scp=77950405026&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950405026&partnerID=8YFLogxK

U2 - 10.1145/1576702.1576721

DO - 10.1145/1576702.1576721

M3 - Conference contribution

AN - SCOPUS:77950405026

SN - 9781605586090

T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

SP - 119

EP - 126

BT - ISSAC 2009 - Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

Y2 - 28 July 2009 through 31 July 2009

ER -