Hi QForms folks,

This week, Lee Troupe will be giving a talk titled “Towards a Function Field Cochrane-Mitchell Theorem.” The abstract is below — hope you see you all there!

-Danny

Abstract: In the initial discussion of Legendre’s Theorem, it was made clear that may be viewed as the n = 3, K = Q case of the celebrated Hasse-Minkowski Theorem.  In general, it is a natural (and wide open!) problem to find other cases of Hasse-Minkowski which can be proved by GoN methods.  One advantage of the GoN approach is that there is also the prospect for an explicit upper bound on the size of an isotropic vector.  (This is also largely an open problem: when explicit bounds are known, they are generally known not to be sharp in any reasonable sense.)

In this talk we pursue the case n = 3, K = F_q(t), where q is an odd prime power.  Both linear forms and Hermite constants have proven analogues in this context.  Perhaps surprisingly, linear forms methods seem more auspicious; in particular, absent a meaningful notion of “majorization” in this context, an approach via Hermite constants doesn’t yield explicit bounds.  We will present a linear forms theorem due to L. Tornheim and try to apply it to prove a Cochrane-Mitchell type result.  This is work (jointly with Pete L. Clark) in progress: as of this writing, it seems that we can prove the result when deg(a), deg(b), deg(c) are neither all even nor all odd.  Help from the audience will be warmly appreciated.