LOCAL NILPOTENCE OF POLYNOMIAL MAP

Abstract

This article discusses polynomial mappings, which are some of the most mysterious objects. One of the tricky problems with polynomial mappings is their reversibility. The main difficulty is the absence of a ring structure in the set of polynomial mappings. The set of polynomial mappings constitutes only a semigroup. Their superposition is considered an operation. The polynomial mappings of A. V. Yagzhev, H. Bass, E. Connel, and D. Wright are considered. In accordance with this mapping, the local nilpotency of the polynomial mapping formed by homogeneous polynomials is shown. This result is related to the fact that the Jacobi matrix of the polynomial map is nilpotent. In this case, the method of matrix multiplication is different from the usual multiplication. Since matrices are variable, their multiplication depends on the points. As the points change, the corresponding Jacobi matrix also changes.