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Register a new userAlgebraic curves in their Jacobian are Sidon sets
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We report new examples of Sidon sets in abelian groups arising from algebraic geometry.
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For each natural number $d$, we introduce the concept of a $d$-cap in $mathbb{F}_3^n$. A subset of $mathbb{F}_3^n$ is called a $d$-cap if, for each $k = 1, 2, dots, d$, no $k+2$ of the points lie on a $k$-dimensional flat. This generalizes the notion of a cap in $mathbb{F}_3^n$. We prove that the $2$-caps in $mathbb{F}_3^n$ are exactly the Sidon sets in $mathbb{F}_3^n$ and study the problem of determining the size of the largest $2$-cap in $mathbb{F}_3^n$.
Our goal is to settle a fading problem, the Jacobian Conjecture $(JC_n)$~: If $f_1, cdots, f_n$ are elements in a polynomial ring $k[X_1, cdots, X_n]$ over a field $k$ of characteristic zero such that $ det(partial f_i/ partial X_j) $ is a nonzero constant, then $k[f_1, cdots, f_n] = k[X_1, cdots, X_n]$. Practically, what we deal with is the generalized one, oindent The Generalized Jacobian Conjecture$(GJC)$ :{it Let $S hookrightarrow T$ be an unramified homomorphism of Noetherian domains. Assume that $S$ is a simply connected UFD ({sl i.e.,} ${rm Spec}(S)$ is simply connected and $S$ is a unique factorization domain) and that $T^times cap S = S^times$. Then $T = S$.} In addition, for consistency of the discussion, we raise some serious (or idiot) questions and some comments about the examples appeared in the papers published by the certain excellent mathematicians (though we are not willing to deal with them). However, the existence of such examples would be against our Main Result above, so that we have to dispute in Appendix B their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples which is shown explicitly.
We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.
We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(lambda_bullet)$ in the orthogonal Grassmannian $mathrm{OG}_n$, which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A. First, using the type B Wronski map, we show that the real locus of the Schubert curve has a natural covering map to $mathbb{RP}^1$, with monodromy operator $omega$ defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute $omega$ without rectifying the skew tableau. The first uses recently-developed shifted tableau crystal operators, while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yongs shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves.
Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain class of dynamical analogues of classical modular curves. The preperiodic points for a quadratic polynomial map may be endowed with the structure of a directed graph satisfying certain strict conditions; we call such a graph admissible. Given an admissible graph $G$, we construct a curve $X_1(G)$ whose points parametrize quadratic polynomial maps -- which, up to equivalence, form a one-parameter family -- together with a collection of marked preperiodic points that form a graph isomorphic to $G$. Building on work of Bousch and Morton, we show that these curves are irreducible in characteristic zero, and we give an application of irreducibility in the setting of number fields. We end with a discussion of the Galois theory associated to the preperiodic points of quadratic polynomials, including a certain Galois representation that arises naturally in this setting.
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