When trying to translate everything that has been done in mathematics to the finite world it is good first to look at the most important theorems. Some of the theorems like the pigeon hole principle or the law of the product or the fundamental theorem of arithmetic do not need to be translated. The fundamental theorem of algebra is different. There is no finite algebraically closed fields for example. When we construct number systems we first look at closing operations, then after introducing a valuation to close it topologically and then close it algebraically. This leads to the number systems . The fundamental theorem of algebra had historically a bit of a rocky start. First stated in 1629 (probably guessed by ignorance by authors like Roth or Lantzenberger) and doubts (as even in 1702 Leibniz thought that can not be factored), a first reasonable proof was only given in 1799 in the doctoral thesis of Gauss (really completed by Ostrowski in 1920) and in 1814 the first solid proof was given by Argand, more proofs were added by Gauss later. Today we have lots of proofs. My personal favorite is the Riemannian geometry proof by Almira and Romero building on the Gauss-Bonnet theorem. The proof I gave in the talk below is based on the Lefschetz fixed point theorem (the proof is mine. I could not find a reference yet, where this is proven as such but its simplicity makes it very likely that it should have been published somewhere. Once you see to look for fixed points rather than roots, it is “obvious”). Equivalent to the fundamental theorem of algebra is the statement:
Theorem: A polynomial map of degree n has n+1 fixed points on the Riemann sphere when counted with multiplicity.
If p(z) is the polynomial. Then the finite roots p(z)=0 are the fixed points of T(z)=p(z)+z. But there is an additional fixed point on the Riemann sphere which has Betti vector (1,0,1). The induced map on cohomology has traces 1 on the 0th cohomology and trace n on harmonic 2-forms (area forms) so that the supertrace on cohomology, the Lefschetz number is n+1. The Lefschetz fixed point theorem assures that the sum of the indices of fixed points is n+1. Because analytic maps have Jacobeans at fixed points that are rotations dilations by Cauchy-Riemann and the fixed point at infinity is stable and also has index 1, all the indices are non-negative. The index of course is the multiplicity of the root. The Lefschetz fixed point theorem immediately gives the result. More even, it also immediately gives Bezout’s theorem which tells that 2 homogeneous polynomials of 3 variables of degree n,m have n*m roots which can be restated that a map on the projective plane that has trace n*m on the volume form cohomlogy, trace 0 on the middle cohomology and so has Lefschetz number n*m+1 must have n*m+1 fixed points. This later formulation for “polynomial map” gives as an assumption of the map the Lefschetz number and so the number of fixed points. But it is this formulation that can be generalized to the finite. The definition of “polynomial map” using cohomology does not really care whether we have an underlying polynomial map. It can for example be deformable to a polynomial map. This deformation does not change the Lefschetz number. One difficulty with deformations however is that the fixed point set can become more complicated making it harder to talk about the “index” in that case. Ini the analytic case, there can not be an accumulation point of fixed points as “analytic” means having Taylor series locally and so heavy rigidity. The Atiyah-Bott fixed point theorem is a possible generalization. Indeed, the heat deformation proof shows that the Lefschetz fixed point statement can be made in any case, where we have a notion of a Dirac operator D and so a Hodge Laplacian and some elliptic regularity so that the kernel of L is finite dimensional so that we have a notion of cohomology.
The finite Lefschetz theorem is much easier because a map on a finite simplicial complex always can only lead to finitely many fixed points on the complex. I proved the Lefschetz fixed point theorem in the finite originally for graph endomorphisms but the statement and proof is the same for simplicial maps, maps of a finite abstract simplicial complex such that T is induced from a map defined on vertices. By looking at the attractor, the statement is almost equivalent to the situation when T is an invertible simplicial map or when the complex comes from a graph (meaning it is the Whitney complex of a graph, which is what I assumed in 2012). Now it was important for the finite Lefschetz fixed point theorem to look at simplices as fixed points, not vertices. Simplices have more structure and allow for a notion of index i(x)= w(x) sign(T|x) where w(x)=-1 for odd dimensional simplices and w(x)=1 for even dimensional simplices. A reflection on the discrete circle for example always has at least 2 fixed points. If n is even then both are vertices or both are edges. If n is odd, one is a vertex and one is an edge. Since edges are reflected and so have sign(T|x)=-1, also w(x)=-1 and the index is still 1. This matches the super trace on cohomology which is 1 on 0-forms and as the trace on 1 forms is -1, the super trace is 1-(-1) = 2 too. In my paper I gave many examples and in particular in the introduction already the special case when G is a complete complex and is a permutation of the vertices. The fixed points are then the finite union of cycles of the permutation. The index of each single cycle is 1, the index of a pair of cycles is -1, the index of a triplet of cycles is 1 again. The sum of all indices is therefore the Euler characteristic of a complete graph where $k$ is the number of cycles. This matches the super trace on cohomology. As is contractible, it has the Betti vector (1,0,0,…,0)$ and the induced map on the 0’th cohomology is 1. The Lefschetz fixed point theorem for an arbitrary permutation on a complete graph is already not totally trivial. I wrote this down in the last paragraph of section 1 in my paper. I only later gave the heat proof. A place I can make out now is a report paper about a HCRP project on differential equations on graphs (Annie Rak steered her research then in a bit different direction and focused on discrete transport equations.)
When modeling polynomial maps which have trace different from 1 on the volume form, we can not use simplicial maps. Polynomial maps must be expanding in part of space as the volume gets multiplied. In the talk I started to try with a more general notion of “map”. One possibility is to allow that vertices can be mapped to simplices or more general to contractible subsimplicial complexes. A contractible subsimplicial complex of G can serve as a “generalized point”. This now has a chance to model maps which multiply the volume form.
[Update of Monday September 8: at the end of the talk I improvised with the map T(x)=2x on (and screwed up a bit). Classically this should model the situation T(x)=2x on the circle T=R/Z. Let us look at the Lefschetz theorem first in this simple classical situation where we have a doubling map on the circle: there is only one fixed point x=0 of course as 2x=x implies x=0 also modulo 1. This fixed point has index -1 as it is unstable (for a one dimensional map, the stable fixed points have index 1 and the unstable fixed points have index -1). The sum of the indices over all fixed points is now -1. The Lefschetz number is also -1 because the trace on the 0-harmonic forms is 1 and the trace on 1-harmonic forms is 2. The super trace is 1-2 =-1. This is like the rotation or reflection case one of the simplest examples that exist for the Lefschetz fixed point theorem.
Now lets look at the discrete case. The only vertex in {0,…,5} that is fixed under the doubling map is v=0. The index of this point is 1. But then there are two edges (5,0) and (0,1) that need to be considered fixed points as they are mapped into something including them. These fixed points have index -1 because their dimension is odd (the induced map is orientation preserving). The total sum of indices is now -1. This is what we want as we want finite mathematics to model classical mathematics (only that things are much easier). As for the super trace of cohomology, also here we have the traace 1 on the harmonic 0-forms and trace 2 on the harmonic 1-forms. The Lefschetz number is again -1.
The same analysis would work on any with large enough n. Note that if we would want to model the map for example, then we would need a discrete circle with more than 2000 vertices. In general, if we want to model a polynomial map on a finite simplicial complex, we need the complex to be large enough to be able to qualify. Axiomatically, we have built the definition of “polynomial map” on the assumption that we can have the right trace induced on area forms. We still need to look at a lot of examples to see whether this all works out nicely.
