ALGEBRAIC VARIETIES OF TWO KELLER POLYNOMIALS IN TWO VARIABLES

Abstract

We study pairs of Keller polynomials in two variables and the associated polynomial mapping

Φ(x,y) = (f(x,y),g(x,y))

under the constant Jacobian condition. We establish that this condition implies strong local geometric constraints on the mapping. In particular, we prove that all fibers of Φ are discrete and locally finite, and admit uniform bounds on compact subsets. We further describe the graph of Φ as a regular surface in R⁴, providing a geometric framework for the analysis. In addition, we derive a differential identity along parametrized rays, which yields an orthogonality relation between the value of the mapping and its differential, together with a non-collinearity constraint under natural nondegeneracy assumptions.