2018/07/29 by Cardinal, Jean-Paul
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.11088
Let f be a polynomial system consisting of n polynomials f1,⋯, fn in n variables x1,⋯, xn, with coefficients in ℚ and let ⟨ f⟩ be the ideal generated by f. Such a polynomial system, which has as many equations as variables is called a square system. It may be zero-dimensional, i.e the system of equations f = 0 has finitely many complex solutions, or equivalently the dimension of the quotient algebra A = ℚ[x]/⟨ f⟩ is finite. In this case, the companion matrices of f are defined as the matrices of the endomorphisms of A, called multiplication maps, xj : \vert h ↦ xjh ., written in some basis of A. We present a practical and efficient method to compute the companion matrices of f in the case when the system is zero-dimensional. When it is not zero-dimensional, then the method works as well and still produces matrices having properties similar to the zero-dimensional case. The whole method consists in matrix calculations. An experiment illustrates the method's effectiveness.