No abstract

We say that two points x, y of a cap C form a free pair of points if any plane containing x and y intersects C in at most three points. For given N and q, we denote by m + 2 (N, q) the maximum number of points in a cap of PG(N, q) that contains at least one free pair of points. It is straightforward to prove that m + 2 (N, q) ≤ (q N−1 + 2q − 3)/(q − 1), and it is known that this bound is sharp for q = 2 and all N . We use geometric constructions to prove that this bound is sharp for all q when N ≤ 4. We briefly survey the motivation for constructions of caps with free pairs of points which comes from the area of statistical experimental design. (2000): 51E22, 62K15 Mathematics Subject Classification

This paper shows how Gröbner basis techniques can be used in coding theory, especially in the construction and decoding of linear codes. A simple algorithm is given for computing the reduced Gröbner basis of the vanishing ideal of a given set of finitely many points, and it is used for finding Padé approximation of any polynomial (given implicitly), which is a major step in decoding. A new method is given for construction of a large class of linear codes that can also be decoded efficiently. These codes include as special cases many of the well known codes such as Reed-Solomon codes, Hermitian codes and, more generally, all one-point algebraic geometry codes.

Gingival fibromatosis (GF) is a condition characterized by a progressive, normal colored enlargement of the gingiva caused by an increase in the size of submucosal connective tissue. Both familial and idiopathic variants of the condition exist. The authors present a case report of a 38-year-old African American man who presented with an impressive overgrowth of the maxillary and mandibular gingivae, subsequently diagnosed as idiopathic GF. In this report, the authors will review the etiologies, treatment, and clinical and histological findings of GF, review similar cases found in the literature, and discuss the differential diagnosis for diffuse gingival enlargements.

Abstract. It is shown how to find general multivariate Padé approximation using the Gröbner basis technique. This method is more flexible than previous approaches, and several examples are given to illustrate this advantage. When the number of variables is small compared to the degree of approximation, the Gröbner basis technique is more efficient than the linear algebra methods in the literature.

A cap in PG(N, q) is said to have a free pair of points if any plane containing that pair contains at most one other point from the cap. In an earlier paper we determined the largest size of caps with free pairs for N = 3 and 4. In this paper we use product constructions to prove similar results in dimensions 5 and 6 that are asymptotically as large as possible. If q > 2 is even, we determine exactly the largest size of a cap in PG(5, q) with a free pair. In PG(5, 3) we give constructions of a maximal size 42-cap having a free pair and of the complete 48-cap that contains it. Additionally, we give some sporadic examples in higher dimensions.