## Mappings of degree 5, part I

### Volume 117 / 2009

#### Abstract

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 $m \geq 4$.