We discuss Bernstein–Walsh type inequalities for holomorphic polynomials restricted to curves of the form z, eP1(z), eP2(z), . . . , ePd (z) ∈ Cd+1, where P1,P2, . . . , Pd are fixed polynomials on C (such that the functions z and ePk(z) are algebraically independent). The existence of such inequalities automatically implies the existence of associated Bernstein–Markov inequalities on the derivatives of polynomials restricted to the curve. The d = 1 case has been much discussed in the recent literature. However, the d > 1 case requires different techniques, and that is the subject of this work.
Polynomial inequalities on exponential curves in $C^n$
BOS, LEONARD PETER;
2010-01-01
Abstract
We discuss Bernstein–Walsh type inequalities for holomorphic polynomials restricted to curves of the form z, eP1(z), eP2(z), . . . , ePd (z) ∈ Cd+1, where P1,P2, . . . , Pd are fixed polynomials on C (such that the functions z and ePk(z) are algebraically independent). The existence of such inequalities automatically implies the existence of associated Bernstein–Markov inequalities on the derivatives of polynomials restricted to the curve. The d = 1 case has been much discussed in the recent literature. However, the d > 1 case requires different techniques, and that is the subject of this work.
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