We extend the big and p-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the p-typical case, it uses positivity with respect to an apparently new basis of the p-typical symmetric functions. We also give...

The Witt construction describes a functor from the category of Rings to the category of characteristic 0 rings. It is uniquely determined by a few associativity constraints which do not depend on the types of the variables considered, in other words, by integer polynomials. This universality allowed Alain Connes and Caterina Consani to devise an analogue of the Witt ring for characteristic one,...

We define the universal 1-adic thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the “dequantization” process and yields Viro’s tropical real hyperfield T R. Then we pr...

For an infinity of number rings we express stable motivic invariants in terms topological data determined by the complex numbers, real and finite fields. We use this to extend Morel's identification endomorphism ring sphere with Grothendieck-Witt quadratic forms deeper base schemes.

In this Note, using an idea due to Thomason [8], we define a “homology theory” on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find Balmers’s higher Witt groups. For more general rings, this homology is isomorphic to the KT-theory of Hornbostel [3], inspired by the work of Williams [9]. ...