As I’ve mentioned before, the number of squarefree monic polynomials of degree n in F_q[t] is exactly q^n – q^{n-1}.

I explained in the earlier post how to interpret this fact in terms of the cohomology of the braid group. But one can also ask whether this identity has a motivic interpretation. Namely: let U be the variety over Q parametrizing monic squarefree polynomials of degree d. So U is a nice open subvariety of affine n-space. Now the identity of point-counts above suggests the question:

**Question: **Is there an identity [U] = [A^n] – [A^{n-1}] in the ring of motives K_0(Var/Q)?

I asked Loeser, who seemed to feel the answer was likely yes, and pointed out to me that one could also ask whether the two classes were identical in the localization K_0(Var/Q)[1/L], where L is the class of A^1. Are these questions different? That is, is there any nontrivial kernel in the natural map K_0(Var/Q) -> K_0(Var/Q)[1/L]? This too is apparently unknown.

Here, I’ll start you off by giving a positive answer in the easy case n=2! Then the monic polynomials are parametrized by A^2, where (b,c) corresponds to the polynomial x^2 + bx + c. The non-squarefree locus (i.e. the locus of vanishing of the discriminant) consists of solutions to b^2 – 4c = 0; the projection to c is an isomorphism to A^1 over Q. So in this case the identity is indeed correct.

**Update**: I totally forgot that Mike Zieve sent me a one-line argument a few months back for the identity |U(F_q)| = q^n – q^{n-1} which is in fact a proof of the motivic identity as well! Here it is, in my paraphrase.

Write U_e for the subvariety of U consisting of degree-d polynomials of the form a(x)b(x)^2, with a,b monic, a squarefree, and b of degree e. Then U is the union of U_e as e ranges from 1 to d/2. Note that the factorisation as ab^2 is unique; i.e, U_e is naturally identified with {monic squarefree polynomials of degree d-2e} x {monic polynomials of degree e.}

Now let V be the space of all polynomials (not necessarily monic) of degree d-2, so that [V] = [A^{n-1}] – [A^{n-2}]. Let V_e be the space of polynomials which factor as c(x)d(x)^2, with d(x) having degree e-1. Then V is the union of V_e as e ranges from 1 to d/2.

Now there is a map from U_e to V_e which sends a(x)b(x)^2 to a(x)(b(x) – b(0))^2, and one checks that this induces an isomorphism between V_e x A^1 and U_e, done.

But actually, now that I think of it, Mike’s observation allows you to get the motivic identity even without writing down the map above: if we write for the space of monic squarefrees of degree d in stratum e, then , and then one can easily compute the class by induction.

### Like this:

Like Loading...