Genera of Numerical Semigroups and Polynomial Identities for Degrees of Syzygies

We derive polynomial identities of arbitrary degree n for syzygies degrees numerical semigroups $$S_m\!=\!\langle d_1,\ldots ,d_m\rangle $$ and show that $$n\ge m$$ they contain higher genera $$G_r\!=\!\sum _{s\in {\mathbb Z}_>\!\!\setminus S_m}s^ r$$ $$S_m$$ . find a number $$g_m\!=\!B_m-m+1$$ algebraically independent $$G_r$$ equations, related any $$g_m+1$$ genera, where $$B_m\!=\! \sum _{k=1}^{m-1}\beta _k$$ $$\beta denote the total partial Betti numbers non-symmetric semigroups. The $$g_m$$ is strongly dependent on symmetry decreases symmetric complete intersections.