TY - GEN
T1 - Bilinear systems with two supports
T2 - 43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018
AU - Bender, Matías R.
AU - Mantzaflaris, Angelos
AU - Faugère, Jean Charles
AU - Tsigaridas, Elias
N1 - Publisher Copyright:
© 2018 Association for Computing Machinery.
PY - 2018/7/11
Y1 - 2018/7/11
N2 - A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix whose elements are the coefficients of the input polynomials up-to sign. This problem is well understood for unmixed multihomogeneous systems, that is for systems consisting of multihomogeneous polynomials with the same support. However, little is known for mixed systems, that is for systems consisting of polynomials with different supports. We consider the computation of the multihomogeneous resultant of bilinear systems involving two different supports. We present a constructive approach that expresses the resultant as the exact determinant of a Koszul resultant matrix, that is a matrix constructed from maps in the Koszul complex. We exploit the resultant matrix to propose an algorithm to solve such systems. In the process we extend the classical eigenvalues and eigenvectors criterion to a more general setting. Our extension of the eigenvalues criterion applies to a general class of matrices, including the Sylvester-type and the Koszul-type ones.
AB - A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix whose elements are the coefficients of the input polynomials up-to sign. This problem is well understood for unmixed multihomogeneous systems, that is for systems consisting of multihomogeneous polynomials with the same support. However, little is known for mixed systems, that is for systems consisting of polynomials with different supports. We consider the computation of the multihomogeneous resultant of bilinear systems involving two different supports. We present a constructive approach that expresses the resultant as the exact determinant of a Koszul resultant matrix, that is a matrix constructed from maps in the Koszul complex. We exploit the resultant matrix to propose an algorithm to solve such systems. In the process we extend the classical eigenvalues and eigenvectors criterion to a more general setting. Our extension of the eigenvalues criterion applies to a general class of matrices, including the Sylvester-type and the Koszul-type ones.
KW - Bilinear system
KW - Determinantal formula
KW - Mixed Multihomogeneous system
KW - Polynomial solving
KW - Resultant
KW - Sparse Resultant
UR - https://www.scopus.com/pages/publications/85054936335
U2 - 10.1145/3208976.3209011
DO - 10.1145/3208976.3209011
M3 - Conference contribution
AN - SCOPUS:85054936335
T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC
SP - 63
EP - 70
BT - ISSAC 2018 - Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation
PB - Association for Computing Machinery
Y2 - 16 July 2018 through 19 July 2018
ER -