Frobenius Number Research Articles

The Frobenius number for shifted geometric sequences associated with the number of solutions

The Frobenius number for shifted geometric sequences associated with the number of solutions

The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number

In this paper we will show that $\MED(F,m)=\{S\mid S \mbox{ is a numeri-}\\ \mbox{cal semigroup with maximal embedding dimension, Frobenius number $F$ and }\\ \mbox{multiplicity }m\}$ is a ratio-covariety. As a consequence, we present two algorithms: one that computes $\MED(F,m)$ and another one that calculates the elements of $\MED(F,m)$ with a given genus. If $X\subseteq S\backslash (\langle m \rangle \cup \{F+1,\rightarrow\})$ for some $S\in \MED(F,m)$, then there exists the smallest element of $\MED(F,m)$ containing $X$. This element will be denoted by $\MED(F,m)[X]$ and we will say that $X$ one of its $\MED(F,m)$-system of generators. We will prove that every element $S$ of $\MED(F,m)$ has a unique minimal $\MED(F,m)$-system of generators and it will be denoted by $\MED(F,m)\msg(S).$ The cardinality of $\MED(F,m)\msg(S)$, will be called $\MED(F,m)$-$\rank$ of $S.$ We will also see in this work, how all the elements of $\MED(F,m)$ with a fixed $\MED(F,m)$-$\rank$ are.

The covariety of numerical semigroups with fixed Frobenius number

Denote by m(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathrm m}(S)$$\\end{document} the multiplicity of a numerical semigroup S. A covariety is a nonempty family C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}$$\\end{document} of numerical semigroups that fulfils the following conditions: there is the minimum of C,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C},$$\\end{document} the intersection of two elements of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}$$\\end{document} is again an element of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}$$\\end{document} and S\\{m(S)}∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\backslash \\{{\\mathrm m}(S)\\}\\in \\mathscr {C}$$\\end{document} for all S∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\\in \\mathscr {C}$$\\end{document} such that S≠min(C).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S\ e \\min (\\mathscr {C}).$$\\end{document} In this work we describe an algorithmic procedure to compute all the elements of C.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}.$$\\end{document} We prove that there exists the smallest element of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {C}$$\\end{document} containing a set of positive integers. We show that A(F)={S∣Sis a numerical semigroup with Frobenius numberF}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {A}(F)=\\{S\\mid S \\hbox { is a numerical semigroup with Frobenius number }F\\}$$\\end{document} is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.

Frobenius Numbers Associated with Diophantine Triples of x2-y2=zr

We give an explicit formula for the p-Frobenius number of triples associated with Diophantine Equations x2−y2=zr (r≥2), that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of Diophantine equations x2−y2=zr. This result is also a generalization because if r=2 and p=0, the (0-)Frobenius number of the Pythagorean triple has already been given. To find p-Frobenius numbers, we use geometrically easy to understand figures of the elements of the p-Apéry set, which exhibits symmetric appearances.

The Covariety of Saturated Numerical Semigroups with Fixed Frobenius Number

In this work, we show that if F is a positive integer, then Sat(F)={S∣S is a saturated numerical semigroup with Frobenius number F} is a covariety. As a consequence, we present two algorithms: one that computes Sat(F), and another which computes all the elements of Sat(F) with a fixed genus. If X⊆S\Δ(F) for some S∈Sat(F), then we see that there exists the least element of Sat(F) containing X. This element is denoted by Sat(F)[X]. If S∈Sat(F), then we define the Sat(F)-rank of S as the minimum of {cardinality(X)∣S=Sat(F)[X]}. In this paper, we present an algorithm to compute all the elements of Sat(F) with a given Sat(F)-rank.

Arithmetic Varieties of Numerical Semigroups

In this paper we present the notion of arithmetic variety for numerical semigroups. We study various aspects related to these varieties such as the smallest arithmetic that contains a set of numerical semigroups and we exhibit the rooted tree associated with an arithmetic variety. This tree is not locally finite; however, if the Frobenius number is fixed, the tree has finitely many nodes and algorithms can be developed. All algorithms provided in this article include their (non-debugged) implementation in GAP.

Wilf inequality is preserved under gluing of semigroups

Wilf Conjecture on numerical semigroups is a question posed by Wilf in 1978 and is an inequality connecting the Frobenius number, embedding dimension and the genus of the semigroup. The conjecture is still open in general. We prove that this Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by [Formula: see text] and [Formula: see text] satisfy the Wilf inequality, then so does their gluing which is minimally generated by [Formula: see text]. We discuss the extended Wilf’s Conjecture in higher dimensions for certain affine semigroups and prove an analogous result.

Ratio-Covarieties of Numerical Semigroups

In this work, we will introduce the concept of ratio-covariety, as a family R of numerical semigroups that has a minimum, denoted by min(R), is closed under intersection, and if S∈R and S≠min(R), then S\{r(S)}∈R, where r(S) denotes the ratio of S. The notion of ratio-covariety will allow us to: (1) describe an algorithmic procedure to compute R; (2) prove the existence of the smallest element of R that contains a set of positive integers; and (3) talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. In addition, in this paper we will apply the previous results to the study of the ratio-covariety R(F,m)={S∣S is a numerical semigroup with Frobenius number F and multiplicitym}.

The Frobenius problem over number fields with a real embedding

The Frobenius problem over number fields with a real embedding

Counting numerical semigroups by Frobenius number, multiplicity, and depth

Counting numerical semigroups by Frobenius number, multiplicity, and depth

The Frobenius number of a family of numerical semigroups with embedding dimension 5

The Frobenius number of a family of numerical semigroups with embedding dimension 5

Teaching perspectives of the Frobenius coin problem of two denominators

Teaching perspectives of the Frobenius coin problem of two denominators

Convexity in (Colored) Affine Semigroups

In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.

The Frobenius problem for numerical semigroups generated by sequences of the form $$ca^n-d$$

The Frobenius problem for numerical semigroups generated by sequences of the form $$ca^n-d$$

The question of Arnold on classification of co-artin subalgebras in singularity theory

In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], Arnold writes: ‘Classification of singularities of curves can be interpreted in dual terms as a description of “co-artin” subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)’. In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let [Formula: see text] be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras [Formula: see text] of the polynomial algebra [Formula: see text] that contains the ideal [Formula: see text] for some [Formula: see text]. It is proven that the set [Formula: see text] is a disjoint union of affine algebraic varieties (where [Formula: see text] is the semigroup of the singularity and [Formula: see text] is the Frobenius number). It is proven that each set [Formula: see text] is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on [Formula: see text]. An isomorphism criterion is given for the algebras in [Formula: see text]. For each algebra [Formula: see text], explicit sets of generators and defining relations are given and the automorphism group [Formula: see text] is explicitly described. The automorphism group of the algebra [Formula: see text] is finite if and only if the algebra [Formula: see text] is not isomorphic to a monomial algebra, and in this case [Formula: see text] where [Formula: see text] is the conductor of [Formula: see text]. The set of orders of the automorphism groups of the algebras in [Formula: see text] is explicitly described.

The Generalized Frobenius Problem via Restricted Partition Functions

Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been generalized: given a nonnegative integer $k$, what is the largest integer with at most $k$ such representations? Other classical values can be generalized too: for example, how many nonnegative integers are representable in at most $k$ ways? For sufficiently large $k$, we give formulas for these values by understanding the level sets of the restricted partition function (the function $f(t)$ giving the number of representations of $t$). Furthermore, we give the full asymptotics of all of these values, as well as reprove formulas for some special cases (such as the $n=2$ case and a certain extremal family from the literature). Finally, we obtain the first two leading terms of the restricted partition function as a so-called quasi-polynomial.

P-Numerical semigroups with p-symmetric properties

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation [Formula: see text] ([Formula: see text] are given positive integers with [Formula: see text]) does not have a non-negative integer solution [Formula: see text]. The generalized Frobenius number (called the [Formula: see text]-Frobenius number) is the largest integer such that this linear equation has at most [Formula: see text] solutions. That is, when [Formula: see text], the [Formula: see text]-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss [Formula: see text]-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer [Formula: see text], [Formula: see text]-gaps, [Formula: see text]-symmetric semigroups, [Formula: see text]-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When [Formula: see text], they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.

A “pseudo-polynomial” algorithm for the Frobenius number and Gröbner basis

Given n⩾2 and a1,…,an∈N. Let S=〈a1,…,an〉 be a semigroup. The aim of this paper is to give an effective pseudo-polynomial algorithm on a1, which computes the Apéry set and the Frobenius number of S. We also find the Gröbner basis of the toric ideal defined by S, for the weighted degree reverse lexicographical order ≺w to x1,…,xn, without using Buchberger's algorithm. As an application we introduce and study some special classes of semigroups. Namely, when S is generated by generalized arithmetic progressions and generalized almost arithmetic progressions with the ratio a positive or a negative number. We determine symmetric and almost symmetric semigroups generated by a generalized arithmetic progression.

Frobenius pseudo-variety of numerical semigroups with a given multiplicity and ratio

AbstractIn this paper, we study the set $$\mathscr {L}(m,r)$$ L ( m , r ) of all numerical semigroups with multiplicity m and ratio r. In particular, we present some algorithms that compute all the elements of $$ \mathscr {L}(m,r)$$ L ( m , r ) with a given genus or with a given Frobenius number.

Bounds for invariants of numerical semigroups and Wilf’s conjecture

Given coprime positive integers g_1< ldots < g_e, the Frobenius number F=F(g_1,ldots ,g_e) is the largest integer not representable as a linear combination of g_1,ldots ,g_e with non-negative integer coefficients. Let n denote the number of all representable non-negative integers less than F; Wilf conjectured that F+1 le e n. We provide bounds for g_1 and for the type of the numerical semigroup S=langle g_1,ldots ,g_e rangle in function of e and n, and use these bounds to prove that F+1 le q e n, where q= Bigg lceil frac{F+1}{g_1} Bigg rceil , and F+1 le e n^2. Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup S=langle g_1,ldots ,g_e rangle is almost-symmetric.