Inequalities between height and deviation of polynomials

Abstract

In this paper, for polynomials with real coefficients P,QP,Q satisfying ∣P(x)∣≤∣Q(x)∣| P\left(x)| \le | Q\left(x)| for each xx in a real interval II, we prove the bound L(P)≤cL(Q)L\left(P)\le cL\left(Q) between the lengths of PP and QQ with a constant cc, which is exponential in the degree dd of PP. An example showing that the constant cc in this bound should be at least exponential in dd is also given. Similar inequalities are obtained for the heights of PP and QQ when the interval II is infinite and P,QP,Q are both of degree dd. In the proofs and in the constructions of examples, we use some translations of Chebyshev polynomials.

Keywords

• height of a polynomial
• chebyshev polynomials
• 11c08
• 12d10
• 26c10
• 41a50