On the one dimensional polynomial and regular images of RnRn

In this work we present a full geometric characterization of the 1-dimensional polynomial and regular images of RnRn. In addition, given a polynomial image S   of RnRn, we compute the smallest positive integer p:=p(S)p:=p(S) such that S   is a polynomial image of RpRp. Analogously, given a regular image S′S′ of RnRn, we determine the smallest positive integer r:=r(S′)r:=r(S′) such that S′S′ is a regular image of RrRr.