Taylor polynomials of rational functions (Q6563076)

Given two polynomials \(P\) and \(Q\) whose constant term is 1, the rational function \(P/Q\) has a Taylor series expansion with constant term 1. Truncating that series at terms of degree \(m\), one obtains the \(m\)th Taylor polynomial of \(P/Q\). A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed.\N\NIn one variable, Taylor varieties are given by rank constraints on Hankel matrices (i.e. matrices which are constant along ascending skew-diagonals from left to right). Inversion of the natural parametrization is known as Padé approximation.\N\NThe authors study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters.\N\NThe authors explain this with Fröberg's Conjecture in commutative algebra.