We prove (Thm. 1.1) that if $e_0>\ldots >e_r>0$ are coprime integers, then the Newton functions $X_1^{e_i}+\ldots +X_r^{e_i}$, $i=0,\ldots ,r$, generate over $\Q$ the field of symmetric rational functions in $X_1,\ldots ,X_r$. This generalizes a previous result of us for $r=2$. This extension requires new methods, including: (i) a study of irreducibility and Galois-theoretic properties of Schur polynomials (Thm. 3.1), and (ii) the study of the dimension of the varieties obtained by intersecting Fermat hypersurfaces (Thm. 4.1). We shall also observe how these results have implications to the study of zeros of linear recurrences over function fields; in particular, we give (Thm. 4.2) a complete classification of the zeros of recurrences of order four with constant coefficients over a function field of dimension $1$.
Newton Functions Generating Symmetric Fields and Irreducibility of Schur Polynomials
DVORNICICH, ROBERTO;
2009-01-01
Abstract
We prove (Thm. 1.1) that if $e_0>\ldots >e_r>0$ are coprime integers, then the Newton functions $X_1^{e_i}+\ldots +X_r^{e_i}$, $i=0,\ldots ,r$, generate over $\Q$ the field of symmetric rational functions in $X_1,\ldots ,X_r$. This generalizes a previous result of us for $r=2$. This extension requires new methods, including: (i) a study of irreducibility and Galois-theoretic properties of Schur polynomials (Thm. 3.1), and (ii) the study of the dimension of the varieties obtained by intersecting Fermat hypersurfaces (Thm. 4.1). We shall also observe how these results have implications to the study of zeros of linear recurrences over function fields; in particular, we give (Thm. 4.2) a complete classification of the zeros of recurrences of order four with constant coefficients over a function field of dimension $1$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
