Greg Smith gave an awesome colloquium here last week about his paper with Blekherman and Velasco on sums of squares.
Here’s how it goes. You can ask: if a homogeneous degree-d polynomial in n variables over R takes only non-negative values, is it necessarily a sum of squares? Hilbert showed in 1888 that the answer is yes only when d=2 (the case of quadratic forms), n=2 (the case of binary forms) or (n,d) = (3,4) (the case of ternary quartics.) Beyond that, there are polynomials that take non-negative values but are not sums of squares, like the Motzkin polynomial
So Greg points out that you can formulate this question for an arbitrary real projective variety X/R. We say a global section f of O(2) on X is nonnegative if it takes nonnegative values on X(R); this is well-defined because 2 is even, so dilating a vector x leaves the sign of f(x) alone.
So we can ask: is every nonnegative f a sum of squares of global sections of O(1)? And Blekherman, Smith, and Velasco find there’s an unexpectedly clean criterion: the answer is yes if and only if X is a variety of minimal degree, i.e. its degree is one more than its codimension. So e.g. X could be P^n, which is the (n+1,2) case of Hilbert. Or it could be a rational normal scroll, which is the (2,d) case. But there’s one other nice case: P^2 in its Veronese embedding in P^5, where it’s degree 4 and codimension 3. The sections of O(2) are then just the plane quartics, and you get back Hilbert’s third case. But now it doesn’t look like a weird outlier; it’s an inevitable consequence of a theorem both simpler and more general. Not every day you get to out-Hilbert Hilbert.
Idle question follows:
One easy way to get nonnegative homogenous forms is by adding up squares, which all arise as pullback by polynomial maps of the ur-nonnegative form x^2.
But we know, by Hilbert, that this isn’t enough to capture all nonnegative forms; for instance, it misses the Motzkin polynomial.
So what if you throw that in? That is, we say a Motzkin is a degree-6d form
for degree-d forms P,Q,R. A Motzkin is obviously nonnegative.
It is possible that every nonnegative form of degree 6d is a sum of squares and Motzkins? What if instead of just Motzkins we allow ourselves every nonnegative sextic? Or every nonnegative homogeneous degree-d form in n variables for n and d less than 1,000,000? Is it possible that the condition of nonnegativity is in this respect “finitely generated?”