11 citations · 60 across the 23 of their papers we have counts for
24 papers
Quartic polynomials in two variables do not represent all non-negative integers
Stanley Yao Xiao, Shuntaro Yamagishi
In this paper, we prove that there does not exist of degree such that . In particular, this answers a question by…
Elliptic curves with a rational 2-torsion point ordered by conductor and the boundedness of average rank
Stanley Yao Xiao
In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion…
Prime values of and , quadratic
Stanley Yao Xiao
We prove an asymptotic formula for primes of the shape with integers and of the shape with prime. Here is a binary quadratic form with integer c…
Heights and quantitative arithmetic on stacky curves
Brett Nasserden, Stanley Yao Xiao
In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satria…
The density of rational points on with three stacky points
Brett Nasserden, Stanley Yao Xiao
In this paper we consider the density of rational points on the "stacky" curve which is with three half points, with res…
On the number of binary quadratic forms having discriminant , prime
Alison Beth Miller, Stanley Yao Xiao
In this paper we obtain an asymptotic formula for the number of -equivalence classes of positive definite binary quadratic forms over $\bZ$ having…
Density of power-free values of polynomials II
Kostadinka Lapkova, Stanley Yao Xiao
In this paper we prove that polynomials of degree , satisfying certain hypotheses, take on the expected density of…
A stratification result for an exponential sum modulo
Kostadinka Lapkova, Stanley Yao Xiao
In this note we consider algebraic exponential sums over the values of homogeneous nonsingular polynomials in the quotient ri…
Uniformity of fibres of period mappings and the -unit equation
Brett Nasserden, Stanley Yao Xiao
In this paper we give a refinement of the method introduced by Lawrence and Venkatesh and thereby showing that their proof of Mordell's conjecture is uniform up to a uniform bound…
On the -Pell equation
Erick Knight, Stanley Yao Xiao
Let , where is a primitive root of unity. In this paper we study the distribution of integers for which the norm equation $N_{K(\sqr…
The negative Pell equation
Erick Knight, Stanley Yao Xiao
By applying methods recently developed by A. Smith with regards to Goldfeld's conjecture, we show that the density of square-free integers in for which the negative Pe…
On monic abelian cubics
Stanley Yao Xiao
In this paper we prove the assertion that the number of monic cubic polynomials with integer coefficients and irreducible, Galois over $\mathbb…
On prime values of binary quadratic forms with a thin variable
Peter Cho-Ho Lam, Damaris Schindler, Stanley Yao Xiao
In this paper we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form with one input restricted to a thin subset of the integers.…
Density of power-free values of polynomials
Kostadinka Lapkova, Stanley Yao Xiao
We establish asymptotic formulae for the number of -free values of polynmilas of degree for any , including w…
Binary quartic forms with vanishing -invariant
Stanley Yao Xiao
We obtain an asymptotic formula for the number of -equivalence classes of irreducible binary quartic forms with integer coefficients with vanishing…
The number of quartic -fields with monogenic cubic resolvent ordered by conductor
Cindy Tsang, Stanley Yao Xiao
In this paper, we consider maximal and irreducible quartic orders which arise from integral binary quartic forms, via the construction of Birch and Merriman, and whose field of fra…
On the representation of integers by binary quadratic forms
Stanley Yao Xiao
In this note we show that for a given irreducible binary quadratic form with integer coefficients, whenever we have for integers , there exists…
Binary quartic forms with bounded invariants and small Galois groups
Cindy Tsang, Stanley Yao Xiao
In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the…
Square-free values of decomposable forms
Stanley Yao Xiao
In this paper we prove that decomposable forms, or homogeneous polynomials with integer coefficients which split completely into linear factors over $\mathbb{…
On the representation of -free integers by binary forms
C. L. Stewart, Stanley Yao Xiao
Let be a binary form with integer coefficients, non-zero discriminant and degree with at least and let denote the largest degree of an irreducible factor of …
On binary cubic and quartic forms
Stanley Yao Xiao
In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic c…
On the representation of integers by binary forms
C. L. Stewart, Stanley Yao Xiao
Let be a binary form with integer coefficients, non-zero discriminant and degree with at least . Let denote the number of integers of absolute value at most…
Zeroes of polynomials with prime inputs and Schmidt's -invariant
Stanley Yao Xiao, Shuntaro Yamagishi
In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions assuming only that the equation satisfies suitable local conditions and the polynomial…
Power-free values of binary forms and the global determinant method
Stanley Yao Xiao
We give an improved estimate for the density of -free values of integral binary forms with no fixed -th power divisor. Further, we give the corresponding improvement to a the…