Solving systems of bivariate algebraic equations by using primitive polynomial remainder sequences
Michael Kalkbrener · 1990
Let K be a field, K the algebraic closure of K and ƒ = qm(x)ym + … + qo(x) a polynomial in K[x,y] with qm ≠ 0. The polynomial qm is called the leading coefficient of ƒ, abbreviated lc(ƒ). The degree of ƒ in y is denoted by deg(ƒ). Let ƒ1, ƒ2, …, ƒk be the primitive polynomial remainder sequence of the primitive polynomials ƒ1 and ƒ2 in K[x,y], abbreviated pprs(ƒ1, ƒ2). For every i ∈ {2, …, k-1} let ci be the content of the pseudoremainder of ƒi-1 and ƒi, li := lc(ƒi)deg(fi-1)-deg(fi)+1, Mi:= {p ∈ K[x] - K | p is irreducible, monic and there exists a j ∈ N such that pj divides c2 · · · ci but not l2 · · · li}, {pi, 1, …, pi, @@@@} := {p ∈ Mi @@@@ p ∉ Mr for r = 2, …, i-1 } and ei := pi,1… pi,@@@@. e2, …, ek-1 is called the elimination sequence of f1 and f2, abbreviated elimseq (ƒ1, ƒ2).