The Center Problem for Analytic Maps with Mixed Even-Degree Terms

We investigate the center problem for analytic maps implicitly defined by algebraic equations of the form F(x,y)=x+y+∑i+k=2nαi,kxiyk=0. Previous studies of homogeneous analytic maps of even degree and of mixed terms of degrees 2 and 4 revealed two recurring algebraic families characterizing the existence of a center at the origin: mirror-symmetry conditions and alternating-sum conditions. In this paper, we resolve the next open case involving mixed terms of degrees 2 and 6 and show that the same algebraic structure extends to mixed terms of degrees 2 and arbitrary even number. Using an approach based on the involutive identity f(f(x))=x, we prove that these two families of algebraic conditions completely characterize the existence of a center at the origin. The proof combines algebraic and structural arguments and avoids the use of explicit higher-order focus quantity computations in the general case.