Let us consider the family of binomial ideals   B=I+J     B  =  I  +  J       , where  J  is lattice ideal and  I  is a square-free quadratic monomial ideal. We give a formula for calculating the Betti numbers of  B . Moreover we bound the Green–Lazarsfeld invariant of a family of quadratic binomial ideals  B  using this formula. This result extends a previous result of Eisenbud et al. for square-free quadratic monomial ideals and extends completely Froberg's theorem. We describe also a subfamily where we can calculate the Green–Lazarsfeld invariant of any ideal  B  and we also compute its first non-linear Betti number.
['Hernan de Alba', 'Marcel Morales']
Betti numbers of binomial ideals