arXiv subjects
Publications and source records attributed to John Yin.
In this note, we fix a height bound $B$, and give a bound on the number of algebraic integers and units of absolute height at most $B$ in cyclotomic extensions $\mathbb{Q}[\zeta_q]$, where $q$ is a prime power. The bound is asymptotic in $q$.
We count certain abelian surfaces with potential quaternionic multiplication defined over a number field $K$ by counting points of bounded height on some genus zero Shimura curves.
We prove the weak form of the generalized Batyrev-Manin-Malle conjecture formulated in \cite{ellenberg2021heights} for the stack $\operatorname{Sym}^2\mathbb{P}^1 := (\mathbb{P}^1 \times \mathbb{P}^1)/S_2$, where the $S_2$ action just permutes the two coordinates. In particular, we show that the diagonal $\Delta \subset \operatorname{Sym}^2\mathbb{P}^1$ is an accumulating substack.
We compute the $p$-adic densities of points with a given splitting type along a (generically) finite map, analogous to the classical Chebotarev theorem over number fields and function fields. Under some mild hypotheses, we prove that these densities satisfy a functional equation in the size of the residue field. This functional equation is a direct reflection of Poincar\'e duality in \'etale cohomology. As a consequence, we prove a conjecture of Bhargava, Cremona, Fisher, and Gajovi\'c on factorization densities of p-adic polynomials. The key tool is the notion of admissible pairs associated to a group, which we use as an invariant of the inertia and decomposition action of a local field on the fibers of the finite map. We compute the splitting densities by M\"obius inverting certain p-adic integrals along the poset of admissible pairs. The conjecture on factorization densities follows immediately for tamely ramified primes from our general results. We reduce the complete conjecture (including the wild primes) to the existence of an explicit "Tate-type" resolution of the "resultant locus" over the integers and complete the proof of the conjecture by constructing this resolution.
We prove that the density of polynomials $P(x)=\sum_{i=0}^n a_n x^n$ over a local field $K$ generating an étale extension with specified splitting type is a rational function in terms of the size of the residue field of $K$ in the case where the splitting type is tame. Moreover, we give a computable recursive formula for these densities and compute the asymptotics of this density as the size of the residue field tends to infinity.
Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the $X_i$ are symmetric matrices of any size, $n\times n$, and enter the formula though tensor product $A_i\otimes X_i$: The solution set of $L_A(X)\succeq0$ is called a free spectrahedron since it contains matrices of all sizes and the defining ``linear pencil" is ``free" of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices $X_i$ of fixed size $n\times n$. The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9\% are also free extreme points. Moreover, the dimension of the active constraint, $\ker(L_A(X^\ell))$, is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples $(X_1^\ell,\dots,X_g^\ell)$. We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points neede
We classify the coefficients $(a_1,...,a_n) \in \mathbb{F}_q[t]^n$ that can appear in a linear relation $\sum_{i=1}^n a_i γ_i =0$ among Galois conjugates $γ_i \in \overline{\mathbb{F}_q(t)}$. We call such an $n$-tuple a Smyth tuple. Our main theorem gives an affirmative answer to a function field analogue of a 1986 conjecture of Smyth over $\mathbb{Q}$. Smyth showed that certain local conditions on the $a_i$ are necessary and conjectured that they are sufficient. Our main result is that the analogous conditions are necessary and sufficient over $\mathbb{F}_q(t)$, which we show using a combinatorial characterization of Smyth tuples due to Smyth. We also formulate a generalization of Smyth's Conjecture in an arbitrary number field that is not a straightforward generalization of the conjecture over $\mathbb{Q}$ due to a subtlety occurring at the archimedean places.