Proof when (m,p)=(2,3)

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2.vi. Proof when (m,p)=(2,3)

As with the case of (m,p)=(2,2), we have a Maple V.5 script which, when run, proves Conjecture 1 in the case when (m,p)=(2,3). Here is the script, and here is its output. (The Maple V.5 script requires two additional scripts, SYMMETRIZE, which computes the primitive part of a symmetrization of a given polynomial and ELEMENTARIZE. which converts symmetric polynomials into polynomials in the elementary symmetric polynomials.)

While the K6/2 300MHz processor we use requires only 14.6 seconds to run the script, it took us several months of computer experimentation to create that script. We originally tried to compute the discriminant in SINGULAR, using standard algorithms to compute the discriminant of a polynomial system. This did not work. Later we found that Maple would easily compute a universal eliminant, and then also the discriminant. (Singular also does this with even more ease.) This polynomial has degree 20 in the 4 parameters s,t,u,v we used and 711 terms. We brazenly tried to show it is positive semidefinite (takes only non-negative values on real parameters) by showing it was a sum of squares. For this, we applied brute force techniques. In the end, we wrote it as a sum of 14 symmetrized squares, in total a sum of 232 squares.

We remark that one cannot hope to write an arbitrary positive semidefinite polynomial of this degree in 4 variables as a sum of squares. The example here may be the largest `naturally occurring' polynomial shown to be a sum of squares. In Section 3.vii we challenge the reader with some larger candidates we have computed.