We study the center problem for polynomial maps y=f(x)=−x−∑n=1∞anxn+1, arising from homogeneous algebraic curves x+y+∑i=0nαn−i,ixn−iyi=x+y+Hn(x,y)=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 for all even degrees n=2k, the center condition is completely characterized by two families of algebraic relations: mirror symmetry conditions and alternating-sum conditions. The proof combines algebraic methods with a direct structural argument. In particular, the necessity part is established without relying on explicit formulas for focus quantities, instead, we make use of the involutive property of the associated map and analyze the symmetric difference Hn(x,f(x))−Hn(f(x),x), which leads to a simple and rigorous characterization of the center condition. This provides a complete and conceptually transparent solution of the homogeneous center problem for polynomial maps.
Petek et al. (Fri,) studied this question.