arXiv subjects

Kazimierz Chomicz

Publications and source records attributed to Kazimierz Chomicz.

4 recordsLinked to original sources

arXiv2026

The type and cardinality of minimal presentations of numerical semigroups with embedding dimension four

For a numerical semigroup $S$ with embedding dimension four, we study the relationship between its type $t(S)$ and the cardinality of its minimal presentations $η(S)$. Using an approach based on the geometry of the Apéry set, we prove that $4t(S) + 5 \geq η(S) \geq t(S)-11$. This resolves a problem of Moscariello and Sammartano asking whether $t(S)$ is bounded by a function of $η(S)$, and improves the previously known bound $9t(S) + 4 \geq η(S)$ due to Bresinsky.

math.CO↗

arXiv2026

On numerical semigroups with embedding dimension four

We develop a geometric procedure for finding the Apéry set of any numerical semigroup with embedding dimension four. Previous methods of comparable strength worked only for embedding dimension three or under very specific conditions. We illustrate our method by finding the Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees of numerical semigroups generated by four consecutive squares and by four consecutive triangular numbers.

math.NT↗

arXiv2025

A Family of Eight-Point Conics Associated with the Cyclic Quadrilateral

We consider the following configuration. Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, and for each vertex $X$, let $H_X$ be the orthocenter of the triangle formed by the other three. Then $A,\;B,\;C,\;D,\;H_A,\;H_B,\;H_C,\;H_D$ all lie on a single conic. In this paper we study a certain generalization of this fact as follows. For an arbitrary point $P_D$ on the Euler line of $\triangle ABC$, we define corresponding points $P_A, P_B, P_C$ on the respective Euler lines such that the ratio $P_XH_X : P_XO$ is constant for all $X$. We show that the four vertices $A,B,C,D$ and the four isogonal conjugates $Q_A,\;Q_B\;,Q_C\;,Q_D$ of the points $P_X$ all lie on a single conic. This result is given distinct treatments, synthetic, projective, and algebraic. Furthermore, we situate the points $P_X$ within the list of triangle centers.

math.MG↗

arXiv2025

On the fourth power level of $\mathfrak{p}$-adic completions of biquadratic number fields

Let $K$ be a number field and $\mathfrak{p} \mid (2)$ be a prime ideal. We compute the fourth level of the $\mathfrak{p}$-adic completions of $K$ when the ramification index is $4$ and the inertial degree is trivial for the ideal $\mathfrak{p}$. This enables the computation of the fourth level of any $\mathfrak{p}$-adic completion of any quartic number field. Here we apply this result to biquadratic number fields and obtain lower bounds for the fourth level of such number fields.

math.NT↗