Highly Degenerate Quadratic Forms over Finite Fields of Characteristic 2

Let K/F be an extension of finite fields of characteristic two. We consider quadratic forms written as the trace of xR(x), where R(x) is a linearized polynomial. We show all quadratic forms can be so written, in an essentially unique way. We classify those R, with coefficients 0 or 1, where the form has a codimension 2 radical. This is applied to maximal Artin-Schreier curves and factorizations of linearized polynomials

Homological stability for classical groups

We prove a slope 1 stability range for the homology of the symplectic, orthogonal and unitary groups with respect to the hyperbolic form, over any fields other than $F_2$, improving the known range by a factor 2 in the case of finite fields. Our result more generally applies to the automorphism groups of vector spaces equipped with a possibly degenerate form (in the sense of Bak, Tits and Wall). For finite fields of odd characteristic, and more generally fields in which -1 is a sum of two squares, we deduce a stability range for the orthogonal groups with respect to the Euclidean form, and a corresponding result for the unitary groups. In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of Galatius-Kupers-Randal-Williams in the case of finite fields, at the characteristic.Comment: v2: Revision. Now recovers the Galatius-Kupers-Randal-Williams improved stability range for general linear groups over finite field

A Higgs Boson Composed of Gauge Bosons

It is proposed to replace the Higgs boson of the standard model by a Lorentz- and gauge-invariant combination of SU(2) gauge bosons. A pair of Higgs bosons is identified with pairs of gauge bosons by setting their mass Lagrangians equal to each other. That immediately determines the mass of the composite Higgs boson. It becomes simply half of the vacuum expectation value of the standard Higgs boson, which matches the observed mass with tree-level accuracy (2%). The two parameters of the standard Higgs potential are replaced by five one-loop self-interactions of the SU(2) gauge bosons, derived from the fundamental gauge couplings. The Brout-Englert-Higgs mechanism of spontaneous symmetry breaking is generalized from scalars to vectors. Their transverse components acquire finite vacuum expectation values which generate masses for both gauge bosons and the Higgs boson. This concept leads beyond the standard model by enabling calculations of the Higgs mass and its potential without adjustable parameters. It can be applied to non-abelian gauge theories in general, such as grand unified models and supersymmetry.Comment: 27 pages, 10 figures, added appendix, fixed errors and typos, clarified the text, added explanation of Equation (15), added reference