On Functional Equations of the Fermat-Waring Type for Non-Archimedean Vectorial Entire Functions

Abstract

Abstract. We show a class of homogeneous polynomials of FermatWaring type such that for a polynomial P of this class, if P (f 1 , . . . , f N+1 ) = P (g 1 , . . . , g N+1 ), where f 1 , . . . , f N+1 ; g 1 , . . . , g N+1 are two families of linearly independent entire functions, then f i = cg i , i = 1, 2, . . . , N + 1, where c is a root of unity. As a consequence, we prove that if X is a hypersurface defined by a homogeneous polynomial in this class, then X is a unique range set for linearly non-degenerate non-Archimedean holomorphic curves.

Lower bounds for genera of fiber products

“…Thus, the functional equation (2), where X, Y are allowed to be entire or meromorphic functions, often studied in the context of Nevanlinna theory (see e.g. [4], [24], [32], [38], [64]), is also related to the low genus problem (see e. g. [7], [33], [44], [45]). Second, algebraic curves (1) with factors of genus zero or one have special Diophantine properties.…”

“…To formulate it explicitly, let us recall that for a holomorphic map between compact Riemann surfaces P : R Ñ C its normalization is defined as a compact Riemann surface N P together with a Galois covering of the lowest possible degree r P : N P Ñ C such that r P " P ˝H for some holomorphic map H : N P Ñ R. From the algebraic point of view, the passage from P to r P corresponds to the passage from the field extension MpRq{P ˚MpCq to its Galois closure. In these terms, the main result of [47] may be formulated as follows: if A and B are rational functions of degree n and m correspondingly such that E A,B is irreducible and gpN A q ą 1, then (4) gpE A,B q ą m ´84n `168 84 .…”