Witt vectors, semirings, and total positivity (Q2827349)

The following section titles will give a fair picture of the content: 1. Introduction, 2. Commutative algebra over \(\mathbb N\), the general, theory, 3. The flat topology over \(\mathbb N\), 4. Plethystic algebra for \(\mathbb N\)-algebras, 5. The composition structure on symmetric functions over \(\mathbb N\), 6. The Schur model for \(\Lambda_\mathbb Z\) over \(\mathbb N\), 7. Witt vectors over \(\mathbb N\)-algebras, 8. Total positivity, 9. A model for the \(p\)-typical symmetric functions over \(\mathbb N\), 10. On the possibility of other models, 11. \(k\)-Schur functions and truncated Witt vectors, 12. Remarks on absolute algebraic geometry. This long paper is worth reading if for no other reason than that the author has made a successful effort to convince the reader that interesting mathematics is to be found at the confluence of Witt vectors, combinatorial positivity and semiring theory. One new result from this territory is as follows: There is a representable comonad on the category of \(\mathbb N\)-algebras which agrees with the \(p\)-typical Witt vector functor \(W_{(p),\infty}\) on \(\mathbb Z\)-algebras. The list of ten open questions adds to the value of the paper. Those who see arithmetic algebraic geometry as semiring theory over \(\mathbb Z\) will certainly find many interesting points of support for that point of view.NEWLINENEWLINEFor the entire collection see [Zbl 1343.14002].