Computing multihomogeneous resultants using straight-line programs

We present a new algorithm for the computation of resultants associated with multihomogeneous (and, in particular, homogeneous) polynomial equation systems using straight-line programs. Its complexity is polynomial in the number of coefficients of the input system and the degree of the resultant computed.

[1]  J. E. Morais,et al.  Straight--Line Programs in Geometric Elimination Theory , 1996, alg-geom/9609005.

[2]  G. B. M. Zerr,et al.  Algebra: 117-118 , 1900 .

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

[4]  Neal H. McCoy.  On the resultant of a system of forms homogeneous in each of several sets of variables , 1933 .

[5]  Joachim von zur Gathen,et al.  Parallel Arithmetic Computations: A Survey , 1986, MFCS.

[6]  Bernd Sturmfels,et al.  Product formulas for resultants and Chow forms , 1993 .

[7]  B. M. Fulk.  MATH , 1992 .

[8]  I. Shafarevich.  Basic algebraic geometry , 1974 .

[9]  J. Sylvester,et al.  XVIII. On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure , 1853, Philosophical Transactions of the Royal Society of London.

[10]  K. Ramachandra,et al.  Vermeidung von Divisionen. , 1973 .

[11]  B. Sturmfels,et al.  Multigraded Resultants of Sylvester Type , 1994 .

[12]  Manfred Minimair.  Sparse Resultant under Vanishing Coefficients , 2003 .

[13]  J. E. Morais,et al.  Lower Bounds for diophantine Approximation , 1996 .

[14]  Ioannis Z. Emiris.  Toric resultants and applications to geometric modelling , 2005 .

[15]  Bernard Mourrain,et al.  Matrices in Elimination Theory , 1999, J. Symb. Comput..

[16]  D. N. Bernshtein.  The number of roots of a system of equations , 1975 .

[17]  Joos Heintz,et al.  Deformation Techniques for Efficient Polynomial Equation Solving , 2000, J. Complex..

[18]  J. Jouanolou,et al.  Le formalisme du résultant , 1991 .

[19]  John F. Canny,et al.  An Efficient Algorithm for the Sparse Mixed Resultant , 1993, AAECC.

[20]  Charles W. Wampler,et al.  A product-decomposition bound for Bezout numbers , 1995 .

[21]  Bernd Sturmfels,et al.  On the Newton Polytope of the Resultant , 1994 .

[22]  C. D'Andrea.  Macaulay style formulas for sparse resultants , 2001 .

[23]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[24]  Joos Heintz,et al.  Testing polynomials which are easy to compute (Extended Abstract) , 1980, STOC '80.

[25]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[26]  Teresa Krick,et al.  The Computational Complexity of the Chow Form , 2002, Found. Comput. Math..

[27]  D. Pedoe,et al.  Methods of Algebraic Geometry, II , 1948 .

[28]  Alicia Dickenstein,et al.  Solving Polynomial Equations , 2005 .

[29]  Andrew McLennan,et al.  The maximum number of real roots of a multihomogeneous system of polynomial equations. , 1999 .

[30]  John F. Canny,et al.  A subdivision-based algorithm for the sparse resultant , 2000, JACM.

[31]  Alicia Dickenstein,et al.  Multihomogeneous resultant formulae by means of complexes , 2003, J. Symb. Comput..

[32]  F. S. Macaulay.  Some Formulæ in Elimination , 1902 .

[33]  C. D'Andrea,et al.  Explicit formulas for the multivariate resultant , 2000, math/0007036.

[34]  Arthur Cayley,et al.  The Collected Mathematical Papers: On the Theory of Elimination , 2009 .

[35]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[36]  bitnetJoos Heintz,et al.  La D Etermination Des Points Isol Es Et De La Dimension D'une Vari Et E Alg Ebrique Peut Se Faire En Temps Polynomial , 1991 .

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