arXiv subjects

Sehun Jeong

Publications and source records attributed to Sehun Jeong.

4 recordsLinked to original sources

arXiv2026

Normal bases of small height in Galois number fields

Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for $K/\mathbb Q$ of bounded Weil height with an explicit bound in terms of the degree and discriminant of $K$. In the case when $d$ is prime, we obtain a particularly good bound using a different method.

math.NT↗

arXiv2024

Integral zeros of quadratic polynomials avoiding sublattices

Assuming an integral quadratic polynomial with nonsingular quadratic part has a nontrivial zero on an integer lattice outside of a union of finite-index sublattices, we prove that there exists such a zero of bounded norm and provide an explicit bound. This is a contribution related to the celebrated theorem of Cassels on small-height zeros of quadratic forms, which builds on some previous work in this area. We also demonstrate an application of these results to the problem of effective distribution of angles between vectors in the integer lattice.

math.NT↗

arXiv2024

Diophantine avoidance and small-height primitive elements in ideals of number fields

Let $K$ be a number field of degree $d$. Then every ideal $I$ in the ring of integers ${\mathcal O}_K$ contains infinitely many primitive elements, i.e. elements of degree $d$. A bound on smallest height of such an element in $I$ follows from some recent developments in the direction of a 1998 conjecture of W. Ruppert. We prove a very explicit bound like this in the case of quadratic fields. Further, we consider primitive elements in an ideal outside of a finite union of other ideals and prove a bound on the height of a smallest such element. Our main tool is a result on points of small norm in a lattice outside of an algebraic hypersurface and a finite union of sublattices of finite index, which we prove by blending two previous Diophantine avoidance results. We also obtain an avoidance result like this for lattice points in the positive orthant in $\mathbb{R}^d$ and use it to obtain a small-height totally positive primitive element in an ideal of a totally real number field outside of a finite union of other ideals. Additionally, we use our avoidance method to prove a bound on the Mahler measure of a generating non-sparse polynomial for a given number field. Finally, we produce a bound on the height of a smallest primitive generator for a principal ideal in a quadratic number field.

math.NT↗

arXiv2017

End-to-End Prediction of Buffer Overruns from Raw Source Code via Neural Memory Networks

Detecting buffer overruns from a source code is one of the most common and yet challenging tasks in program analysis. Current approaches have mainly relied on rigid rules and handcrafted features devised by a few experts, limiting themselves in terms of flexible applicability and robustness due to diverse bug patterns and characteristics existing in sophisticated real-world software programs. In this paper, we propose a novel, data-driven approach that is completely end-to-end without requiring any hand-crafted features, thus free from any program language-specific structural limitations. In particular, our approach leverages a recently proposed neural network model called memory networks that have shown the state-of-the-art performances mainly in question-answering tasks. Our experimental results using source codes demonstrate that our proposed model is capable of accurately detecting simple buffer overruns. We also present in-depth analyses on how a memory network can learn to understand the semantics in programming languages solely from raw source codes, such as tracing variables of interest, identifying numerical values, and performing their quantitative comparisons.

cs.SE↗