Related papers: Heights of mixed motives
We introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve. We discuss the modularity of an abelian variety over the rational number field. We…
The second author proved that the set of post-critically finite polynomials of given degree is a set of bounded height, up to change of variables. Motivated by an observation about unicritical polynomials, we complement this by proving that…
Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a…
The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.
We prove asymptotic formulas for the number of rational points of bounded height on certain blow-ups of the projective space.
We regard explanations as a blending of the input sample and the model's output and offer a few definitions that capture various desired properties of the function that generates these explanations. We study the links between these…
We give uniform upper bounds for the number of rational points of height at most $B$ on non-singular complete intersections of two quadrics in $\mathbb{P}^3$ defined over $\mathbb{Q}$. To do this, we combine determinant methods with descent…
The purpose of this paper is to give a linear and effective height inequality for algebraic points on curves over functional fields. Our height inequality can be viewed as the logarithmic canonical class inequality of a punctured curve over…
We argue that a certain distribution of matter in higher dimensions can provide the correct behaviour of gravity in four dimensions. Some explicit examples illustrating the idea are considered.
This paper investigates the structure of generic motives and their implications for the motivic cohomology of fields. Originating in Voevodsky's theory of motives and related to Beilinson's vision of a motivic $t$-structure, generic motives…
In this paper we suggest a definition for the category of mixed motives generated by the motive h^1(E) for E an elliptic curve without complex multiplication. We then compute the cohomology of this category. Modulo a strengthening of the…
The purpose of this paper is to describe the images of multilinear polynomials of arbitrary degree on the strictly upper triangular matrix algebra.
We characterize the mixed discriminant of positive semi definite matrices using its most basic properties. As a corollary we establish its minimality among non negative and multi additive functionals.
There is a parallelism between growth in arithmetic combinatorics and growth in a geometric context. While, over $\mathbb{R}$ or $\mathbb{C}$, geometric statements on growth often have geometric proofs, what little is known over finite…
We compute the number of points over finite fields of some algebraic varieties related to cluster algebras of finite type. More precisely, these varieties are the fibers of the projection map from the cluster variety to the affine space of…
The dimensions of certain varieties defined by monomials are computed using only high school algebra.
We show the existence of canonical heights of subvarieties for bounded sequences of morphisms and give some applications.
We define a notion of colimit for diagrams in a motivic category indexed by a presheaf of spaces (e.g. an \'etale classifying space), and we study basic properties of this construction. As a case study, we construct the motivic analogs of…
Let $L$ be a nef and big line bundle on a scheme $X$. It is well known that if $X$ is a projective over a field then the augmented base locus and the exceptional base locus agree. This result is extended to projective schemes over arbitrary…
We give a more coherent definition of upward planar order.