Detection of special curves via the double resultant
Given a polynomial of degree \(n\), \(F=a_ny^n+\ldots+a_0\), \(a_n\neq 0\), with coefficients in a commutative ring, the discriminant is equal, up to sign, to the quotient of the resultant of \(F\) and \(dF/dy\) by the leading coefficient \(a_n\). It is well known that a nonconstant polynomial over a field has a multiple root if and only if its discriminant is zero. In literature, it is frequently used the double (or iterated) discriminant of multivariate polynomials in order to deal with different problems of algebraic geometry. In this paper, the authors analyze the use of the double resultant instead. Given a bivariate polynomial whose coefficients depend on a parameter \(b\), \(F_b(x,y)\), the double resultant is applied in order to compute the values of \(b\) for which \(F_b(x,y)\) has singularities. Some examples of different nature where the double resultant is used are introduced: special level sets of rational first integrals for rational discrete dynamical systems; remarkable values of rational first integrals of polynomial vector fields; bifurcation values in phase portraits of polynomial vector fields; and the different topologies of the offset of curves.
scientific article; zbMATH DE number 6855267
scientific article