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Ricardo Conceição

Publications and source records attributed to Ricardo Conceição.

8 recordsLinked to original sources
arXiv2023

A polynomial analogue of Berggren's theorem on Pythagorean triples

Say that $(x, y, z)$ is a positive primitive integral Pythagorean triple if $x, y, z$ are positive integers without common factors satisfying $x^2 + y^2 = z^2$. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on $(3, 4, 5)$ and $(4, 3, 5)$ generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren's theorem in the context of a one-variable polynomial ring over a field of characteristic $\neq 2$. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.

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arXiv2020

The Markoff equation over polynomial rings

When $A=3$, the positive integral solutions of the so-called Markoff equation $$M_A:x^2 + y^2 + z^2 = Axyz$$ can be generated from the single solution $(1,1,1)$ by the action of certain automorphisms of the hypersurface. Since Markoff's proof of this fact, several authors have showed that the structure of $M_A(R)$, when $R$ is $\mathbf{Z}[i]$ or certain orders in number fields, behave in a similar fashion. Moreover, for $R=\mathbf{Z}$ and $R=\mathbf{Z}[i]$, Zagier and Silverman, respectively, have found asymptotic formulae for the number of integral points of bounded height. In this paper, we investigate these problems when $R$ is a polynomial ring over a field $K$ of odd characteristic. We characterize the set $M_A(K[t])$ in a similar fashion as Markoff and previous authors. We also give an asymptotic formula that is similar to Zagier's and Silverman's formula.

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arXiv2020

On integral points on isotrivial elliptic curves over function field

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note, we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$, the size of $S$ and the rank of the Mordell-Weil group $E(L)$. In the second part, we assume that $L$ is the function field of a hyperelliptic curve $C_A:s^2=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_A$, then we prove that the set of separable $\{\infty\}$-points can be bounded solely in terms of $g$ and does not seem to depend on the Mordell-Weil group $E(L)$. This is done by bounding the number of separable integral points over $k(t)$ on elliptic curves of the form $E_A:A(t)y^2=f(x)$, where $f(x)$ is a polynomial over $k$. Additionally, we show that, under an extra condition on $A(t)$, the existence of a separable integral point of "small" height on the elliptic curve $E_A/k(t)$ determines the isomorphism class of the elliptic curve $y^2=f(x)$.

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arXiv2014

On a Frobenius problem for polynomials

We extend the famous diophantine Frobenius problem to the case of polynomials over a field $k$. Similar to the classical problem, we show that the $n=2$ case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over $\mathbb{Z}$ to $k[t]$ and give an algorithm to solve the Frobenius problem for polynomials over a field $k$ of sufficiently large size.

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arXiv2013

Explicit points on the Legendre curve II

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$ is given in terms of an elementary property of the subgroup of $(\Z/d\Z)^\times$ generated by $p$. We show that for many values of $d$ the rank is large. For example, if $d$ divides $2(p^f-1)$ and $2(p^f-1)/d$ is odd, then the rank is at least $d/2$. When $d=2(p^f-1)$, we exhibit explicit points generating a subgroup of $E(\Fq(t^{1/d}))$ of finite index in the "2-new" part, and we bound the index as well as the order of the "2-new" part of the Tate-Shafarevich group.

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arXiv2012

Elliptic curves over function fields with a large set of integral points

We construct isotrivial and non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type varieties over $\mathbb_{F}_q(t)$ with a Zariski dense set of separable integral points. This provides a counterexample to a natural translation of the Lang-Vojta conjecture to the function field setting. We also show that our main result provides examples of elliptic curves with an explicit and arbitrarily large set of linearly independent points.

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arXiv2011

On the characterization of minimal value set polynomials

We give an explicit characterization of all minimal value set polynomials in $\F_q[x]$ whose set of values is a subfield $\F_{q'}$ of $\F_{q}$. We show that the set of such polynomials, together with the constants of $\F_{q'}$, is an $\F_{q'}$-vector space of dimension $2^{[\F_{q}:\F_{q'}]}$. Our approach not only provides the exact number of such polynomials, but also yields a construction of new examples of minimal value set polynomials for some other fixed value sets. In the latter case, we also derive a non-trivial lower bound for the number of such polynomials.

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