Department of Mathematics

Given a generic choice of polynomials with complex coefficients, one can compute the dimension of the straight lines, planes, or d-dimensional planes contained in the common zeros of the polynomials. When this dimension is 0, there is some finite number of d-planes. For example, there are 321,489 3-dimensional planes in the zero locus of a degree 3 homogeneous polynomial in 9 variables over the complex numbers. This number can be identified with the topological Euler number of a certain vector bundle. However, it only corresponds to the count of d-planes over an algebraically closed field like the complex numbers. We can get information over other fields like the real numbers, the rational numbers, or finite fields, by using an Euler number from A1-homotopy theory instead. This Euler number is no longer an integer; instead it is a bilinear form, and invariants of bilinear forms record information about the arithmetic and geometry of the planes. In this talk, we will introduce these enumerative problems and A1-Euler numbers. We establish integrality results for the A1-Euler class, and use this to compute the Euler numbers associated to arithmetic counts of d-planes on complete intersections in terms of topological Euler numbers over the real and complex numbers. The example in the title then follows from work of Finashin–Kharlamov. The new work in this talk is joint with Tom Bachmann.

(Contact Person: Anna Marie Bohmann)