The Center Problem for Homogeneous Case of Polynomial Maps

We study the center problem for polynomial maps y=f(x)=−∑n=0∞anxn+1, arising from homogeneous algebraic curves x+y+∑k=0nαn−k,kxn−kyk=0. While explicit conditions were previously known only for low even degrees n = 2,4,6,8,10, their general structure remained conjectural. In this paper we resolve the case n = 12 and prove that the observed algebraic patterns completely characterize the center for all even degrees n. More precisely, we show that the center condition is equivalent to one of two explicit families of algebraic relations. This provides a complete classification of the center problem in the homogeneous case.