The d th linear polarization constant of RdRd

In [C. Benítez, Y. Sarantopoulos, A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (3) (1998) 395–408] it was conjectured that for all unit vectors u1,…,udu1,…,ud in RdRd,X(u1,…,ud):=supx∈Rd,|x|2=d∏i=1d〈x,ui〉2⩾1 with equality occurring iff u1,…,udu1,…,ud are orthonormal. We relate this to a conjecture about solutions of Ay=y−1Ay=y−1, where A=(〈ui,uj〉)A=(〈ui,uj〉), and show that if the conjecture fails then the u1,…,udu1,…,ud minimizing XX must be linearly dependent. We also show X(u1,…,ud)⩾1X(u1,…,ud)⩾1 for certain families of u1,…,udu1,…,ud.