Intercalate matrices and the Yuzvinsky Conjecture

The property |z||w| = |zw| of complex numbers leads naturally to the following question concerning sums of squares: in the polynomial ring R = Z[x_1,...,x_r, y_1,... y_s], what is the minimal n such that [(x_1)^2 + ... +(x_r)^2] [(y_1)^2 + ... +(y_s)^2] can be expressed as a sum of n squares?

Given any such expression, one can set up an r x s matrix M and paint its entries using n available colors. This coloured M has certain intercalation properties which I shall explain in my talk.

To tackle the sums of squares question one wold like to know, for given r and s, the chromatic number of r x s intercalate matrices. In the 1980’s S.Yuzvinsky put forth a conjecture on this chromatic number. I shall present his conjecture as an open problem, together with a partial solution.