Factoring analytic multivariate polynomials and non-standard Cauchy-Riemann conditions

Abstract

Motivated by previous work on the simplification of parametrizations of curves, in this paper we generalize the well known notion of analytic polynomial (a bivariate polynomial $P(x,y)$, with complex coefficients, which arises by substituting $z \rightarrow x+\ii y$ on a univariate polynomial $p(z)\in \mathbb{C}[z]$, i.e. $p(z)\rightarrow p(x+\ii y)=P(x, y)$) to other finite field extensions, beyond the classical case of $\mathbb{R}\subset \mathbb{C}$. In this general setting we obtain different properties on the factorization, {\it gcd}'s and resultants of analytic polynomials, which seem to be new even in the context of Complex Analysis. Moreover, we extend the well-known Cauchy-Riemann conditions (for harmonic conjugates) to this algebraic framework, proving that the new conditions also characterize the components of generalized analytic polynomials.