I have a small, self-contained, math question, whose motivation is from theoretical computer science (specifically, list decoding of algebraic codes, derivative/multiplicity codes, etc).I wonder whether someone might have an idea.I'm looking for an operator T that can be applied to m-variate polynomials over a finite field.When applied to a polynomial, the operator should yield a polynomial of not much higher degree. The operator should satisfy the following property:for every m-variate polynomial $F$, and m univariate non-constant polynomials $g_1,...,g_m$, if you know $g_1,...,g_m$ and $F(g_1(t),...,g_m(t))$ for a parameter t, then you can also compute $TF(g_1(t),...,g_m(t))$. The operator should be non-trivial, in the sense that for a fixing $x_1,..,x_m$, the value $F(x_1,...,x_m)$ does not determine the value of $TF(x_1,...,x_m)$. For $m=1$, the derivative operator gives exactly that:By the chain rule$(F(g(t)))' = F'(g(t))\cdot g'(t)$, which implies that$F'(g(t)) = (F(g(t)))'/g'(t)$.My question is whether there is an operator that works for general m.
Looking for an operator on polynomials
it.information theory;coding theory;algebra;polynomials