A Negative Answer to a Question of Basu and Perrucci on Two Multi-Affine Polynomials

A real polynomial is multi-affine if it has degree at most one in each variable. Basu and Perrucci proved that the real zero set of one multi-affine polynomial of degree d in R^n has at most 2^(d-1) connected components, independently of n. They asked whether the number of connected components of the common real zero set of two multi-affine polynomials of degree at most d is bounded in terms of d alone; the question was posed again in Oberwolfach Report 9/2025. We show that the answer is no. For m >= 2, explicit multi-affine polynomials of degrees 2 and 4 in 2m+1 variables have a common real zero set homeomorphic to a product of m hyperbolas, with exactly 2^m connected components. A sum-of-squares identity gives a short exact certificate. The construction also combines multi-affine polynomials in disjoint variable sets into a pair whose zero set is homeomorphic to the product of their zero sets. The negative answer persists inside boxes and, as a lower bound, inside any set with nonempty interior.