Spaces of Polynomials Related to Multiplier Maps

Abstract

Let f (x) be a complex polynomial of degree n. We attach to f a C-vector space W (f ) that consists of complex polynomials p(x) of degree at most n − 2 such that f (x) divides f ′′ (x)p(x) − f ′ (x)p ′ (x). W (f ) originally appears in Yuri Zarhin's solution towards a problem of dynamics in one complex variable posed by Yu. S. Ilyashenko. In this paper, we show that W (f ) is nonvanishing if and only if q(x) 2 divides f (x) for some quadratic polynomial q(x). Then we prove W (f ) has dimension (n − 1) − (n 1 + n 2 + 2N 3 ) under certain conditions, where n i is the number of distinct roots of f with multiplicity i and N 3 is the number of distinct roots of f with multiplicity at least three. Definitions, notation, and statementsWe write C for the field of complex numbers, C[x] for the ring of one variable polynomials with complex coefficients, and P s ⊆ C[x] for the subspace of polynomials of degree at most s. All vector spaces we consider are over C unless otherwise stated. Throughout the paper fand N 3 is the number of distinct roots of f with multiplicity at least three.We are interested in the space W (f ) ⊆ P n−2 , for which every p(x) ∈ W (f ) satisfies the condition that p(x) divides f ′′ (x)p(x) − f ′ (x)p ′ (x). As a subspace of P n−2 , we wish to compute the dimension of W (f ) for various f (x). The following assertions are main results of this paper.(iii) If n 1 = r = 4, and f (x) has at least two distinct multiple roots, then dim[W (f )] = 2 > µ = 1.Note that by rewriting n asUsing the lower bound of dim[W (f )] in Theorem 1.1.(i), we deduce if W (f ) vanishes then r ≤ n 1 − 1. Combining with the above expression of r, we obtainSince n i are nonnegative integers, we deduce from the above inequality that (n 2 , n 3 ) ∈ {(0, 0), (0, 1), (1, 0)} and n i = 0 for all i ≥ 4. This condition is equivalent to saying that for all quadratic polynomials q(x) ∈ C[x], q(x) 2 does not divide f (x). So by taking the contrapositive, we prove the following corollary.Corollary 1.2. If there exists a quadratic q(x)In fact, Zarhin showed that [6, Theorem 1.5.(ii)] the nonvanishing condition of W (f ) in Corollary 1.2 is sufficient. Suppose f is monic with distinct roots α 1 , . . . , α n . We consider a holomorphic map M : P n → C n defined via f (x) −→ (f ′ (α 1 ), . . . , f ′ (α n )). At first, Zarhin used W (f ) to show [6, Theorem 1.1] that the rank of the tangent map dM is n − 1 at all points of P n . Based on these results, Zarhin proved [7, Theorem 1.6] that the multipliers of any n − 1 distinct periodic orbits, considered as multi-valued algebraic maps on P n , are algebraically independent over C under certain conditions. This question about algebraic independence was asked by Yu. S. Ilyashenko in the context of studying the Kupka-Smale property for volume-preserving polynomial automorphisms of C 2 . (see [1,3] for a detailed discussion).