1. CMB 2010 (vol 54 pp. 39)
|Elements in a Numerical Semigroup with Factorizations of the Same Length|
Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.
Keywords:numerical monoid, numerical semigroup, non-unique factorization
Categories:20M14, 20D60, 11B75
2. CMB 2003 (vol 46 pp. 229)
|Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials|
|Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials |
Recently there has been tremendous interest in counting the number of integral points in $n$-dimen\-sional tetrahedra with non-integral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in $n$-dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous $n$-dimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting.
Categories:11B75, 11H06, 11P21, 11Y99