Isolated Real Zeros of Sums of Squares: Quaternary Forms and Quinary Quartics

For even n let B′_{n,m} be the supremum of the number of real projective zeros of a sum of squares of real forms of degree n in m variables, taken over those with finitely many real zeros. Fröberg, Lundqvist, Oneto and Shapiro conjectured that B′_{n,m} = (n/2)^{m−1}; Choi, Lam and Reznick had proved the lower bound and the case m = 3. A related conjecture of Ottaviani and Shapiro states that a sum of squares of real polynomials of degree at most k in l variables has at most k^l isolated real zeros; it was known for l ≤ 2. Both conjectures follow from the statement that the real points of the base locus of a real linear system of forms of degree k on ℙ^N have at most k^N isolated points. We prove this statement for N = 3 and every k, and for k = 2, N = 4. Consequently B′_{2k,4} = k^3 for every k and B′_{4,5} = 16, and the Ottaviani–Shapiro bound holds for l = 3 and every k and for (k,l) = (2,4). We also give a short proof of B′_{4,4} = 8, a value that Choi, Lam and Reznick expected and that Dressler stated without proof. The main tools are a weighted Bézout inequality, a Jacobian argument along the curve components of the base locus, and an integral closure estimate for the system of partial derivatives. Both conjectures remain open in general, for instance for sextics in five variables and for quartics in six variables. This is an unrefereed note.