## Spec is representable

Saw Matt Baker at the Joint Meetings and he told me about this crazy paper he just posted, “Matroids over Hyperfields.”   A hyperring is just like a ring except addition is multivalued; given elements x and y of R, x+y is a subset of R which you can think of as “the possible outcomes of summing x and y.”  A hyperfield is a hyperring in which every nonzero element has a multiplicative inverse.

Here’s an example familiar to tropical geometers:  let T be the hyperfield whose elements are $\mathbf{R} \bigcup -\infty$, whose multiplication law is real addition, and whose addition law is

a + b = max(a,b) if a <> b

a + b = {c: c < a} if a=b

In other words, each element of T can be thought of as the valuation of an otherwise unspecified element of a field with a non-archimedean valuation, and then the addition law answers the question “what is ord(x+y) if ord(x) = a and ord(y) = b”?

This may sounds at first like an almost aggressively useless generalization, but no!  The main point of Matt’s paper is that it makes sense to talk about a matroid with coefficients in a hyperfield, and that lots of well-studied flavors of matroids can be written as “matroids over F” for a suitable hyperfield F; in this way, a lot of different stories about different matroid theories get unified and cleaned up.

In fact, a matroid itself turns out to be the same thing as a matroid over K, where K is the Krasner hyperfield:  just two elements 0 and 1, with the multiplication law you expect, and addition given by

0 + 0 = 0

0 + 1 = 1

1 + 1 = {0,1}

One thing I like about K is that it repairs the problem (if you see it as a problem) that the category of fields has no terminal object.  K is terminal in the category of hyperfields; any hyperfield (and in particular any field) has a unique map to K which sends 0 to 0 and everything else to 1.

More generally, as Matt observes, if R is a commutative ring, a homomorphism f from R to K is nothing other than a prime ideal of R — namely, f^{-1}(0).  So once you relax a little and accept the category of hyperfield, the functor Spec: Rings -> Sets is representable!  I enjoy that.

Update:  David Goss points out that this observation about Spec and the Krasner hyperfield is due to Connes and Consani in “The hyperring of adèle classes” JNT 131, (2011) 159-194, p.161.  In fact, for any scheme X of finite type over Z, the underlying Zariski set of X is naturally identified with Hom(Spec(K),X); so Spec(K) functions as a kind of generic point that’s agnostic to characteristic.

## What I learned at the Joint Math Meetings

Another Joint Meetings in the books!  My first time in San Antonio, until last weekend the largest US city I’d never been to.  (Next up:  Jacksonville.)  A few highlights:

• Ngoc Tran, a postdoc at Austin, talked about zeroes of random tropical polynomials.  She’s proved that a random univariate tropical polynomial of degree n has about c log n roots; this is the tropical version of an old theorem of Kac, which says that a random real polynomial of degree n has about c log n real roots.  She raised interesting further questions, like:  what does the zero locus of a random tropical polynomial in more variables look like?  I wonder:  does it look anything like the zero set of a random band-limited function on the sphere, as discussed by Sarnak and Wigman?  If you take a random tropical polynomial in two variables, its zero set partitions the plane into polygons, which gives you a graph by adjacency:  what kind of random graph is this?