Weights of exact threshold functions

We consider Boolean exact threshold functions defined by linear equations and, more generally, polynomials of degree $d$. We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close. In the linear case in particular they are almost matching. This quantity is the same as the maximum magnitude of the integer coefficients of linear equations required to express every possible intersection of a hyperplane in $\mathbb R^n$ and the Boolean cube $\{0,1\}^n$ or, in the general case, intersections of hypersurfaces of degree $d$ in $\mathbb R^n$ and $\{0,1\}^n$. In the process we construct new families of ill-conditioned matrices. We further stratify the problem (in the linear case) in terms of the dimension $k$ of the affine subspace spanned by the solutions and give upper and lower bounds in this case as well. There is a substantial gap between these bounds, a challenge for future work.