ALGEBRAIC VARIETIES OF TWO KELLER POLYNOMIALS IN TWO VARIABLES (Journal of Mathematics Mechanics and Computer Science, 2026)
We study pairs of Keller polynomials in two variables and the associated polynomial mappingΦ(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.
Keywords: Keller polynomial; polynomial automorphism; Jacobian conjecture; algebraic variety; two variables