Optimal bounds for sign-representing the intersection of two halfspaces by polynomials

The threshold degree of a function f:{0,1}^n->{-1,+1} is the least degree of a real polynomial p with f(x)=sgn p(x). We prove that the intersection of two halfspaces on {0,1}ⁿ has threshold degree Omega(n), which matches the trivial upper bound and completely answers a question due to Klivans (2002). The best previous lower bound was Omega(sqrt n). Our result shows that the intersection of two halfspaces on {0,1}ⁿ only admits a trivial 2^{{Theta(n)}-time} learning algorithm based on sign-representation by polynomials, unlike the advances achieved in PAC learning DNF formulas and read-once Boolean formulas. The proof introduces a new technique of independent interest, based on Fourier analysis and matrix theory.