An algorithm for implicit interpolation

Abstract

We study the multivariate polynomial interpolation problem when the set of nodes is a zero-dimensional affine variety $V \subset {\mathbb {C}}^n$ given as the set of zeros of polynomials $F_1, \ldots , F_n$ with integer coefficients that generate a radical ideal. We describe a symbolic procedure to construct, from $F_1,\ldots ,F_n$, a basis $\{G_1, \ldots , G_D\}$ of a space of interpolants $\varPi _V$ for $V$, where $D$ is the number of nodes. This construction yields a space of interpolants $\varPi _V$ which is uniquely determined by the sequence $F_1,\ldots ,F_n$ and such that the degree of the interpolants is at most $n(d-1)$, where $d$ is an upper bound for the degrees of $F_1, \ldots , F_n$. Furthermore, we exhibit a probabilistic algorithm that, from $F_1, \ldots , F_n$ and a given additional polynomial $F$, computes the polynomials $G_1, \ldots , G_D$ and the interpolant $P_F$ of $F$ with roughly ${\fancyscript{O}}^{\sim }(L\delta ^3h\deg (F)h(F)h(V))$ bit operations. Here $L$ is the cost of evaluation of $F_1, \ldots , F_n$ and $F,\,\delta$ the degree of the input system $F_1,\ldots ,F_n,\,h$ an upper bound for the heights of the polynomials $F_1,\ldots ,F_n,\,h(V)$ the height of $V,\,\deg (F)$ the degree of $F$, and $h(F)$ the height of $F$. The numbers $\delta$ and $h(V)$ are always bounded by $d^n$ and $nh+2n\log (n+1)d^n$ respectively and in certain cases of practical interest these numbers are considerably smaller than these bounds.