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

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 → x+ iy on a univariate polynomial p(z) ∈ C[z], i.e. p(z)→ p(x+ iy) = P (x, y)) to other finite field extensions, beyond the classical case of R ⊂ C. In this general setting we obtain different properties on the factorization, 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.