Mappings of degree 5, part I

M. Maciejewski ¹   ; A. Prószyński ¹

¹ Kazimierz Wielki University 85-072 Bydgoszcz, Poland

Colloquium Mathematicum, Tome 117 (2009) no. 2, pp. 223-242

Résumé

The class of linear (resp. quadratic) mappings over a commutative ring is determined by a set of equation-type relations. For the class of homogeneous polynomial mappings of degree $m \geq 3$ it is so over a field, and over a ring there exists a smallest equationally definable class of mappings containing the preceding one. It is proved that generating relations determining that class can be chosen to be strong relations (that is, of the same form over all commutative rings) if{f} $m \leq 5$. These relations are already found for $m \leq 4$. The purpose of the present paper is to find the first of two parts of generating relations (namely, all the 3-covering relations) satisfied by homogeneous polynomial mappings of degree 5. Moreover, we find some strong $(m-2)$-relations for any degree $m \geq 4$.

Détail

DOI : 10.4064/cm117-2-5

Keywords: class linear resp quadratic mappings commutative ring determined set equation type relations class homogeneous polynomial mappings degree geq field ring there exists smallest equationally definable class mappings containing preceding proved generating relations determining class chosen strong relations form commutative rings leq these relations already found leq purpose present paper first parts generating relations namely covering relations satisfied homogeneous polynomial mappings degree moreover strong m relations degree geq