New inequalities for the extended Euler-Poincar\'{e} theorem

In 1988, Bj\"{o}rner and Kalai [Acta Math. 161 (3--4) (1988) 279--303] extended the classic Euler-Poincar\'{e} theorem by introducing certain nonlinear relations between the $f$-vector and the Betti sequence of simplicial complexes. In this paper, we present an equivalent characterization of Bj\"{o}rner and Kalai's nonlinear relations using a number-theoretic approach. Moreover, we strengthen a result of Bj\"{o}rner and Kalai concerning the maximal element of Betti sequences with respect to a fixed $f$-vector and the minimal element of $f$-vectors with respect to a fixed Betti sequence.