Bernstein–Walsh Inequalities in Higher Dimensions over Exponential Curves

Let x= (x₁, ⋯, x_d) ∈ [- 1 , 1] ^d be linearly independent over Z, and set K={(ez,ex1z,ex2z⋯,exdz):|z|≤1}. We prove sharp estimates for the growth of a polynomial of degree n, in terms of En(x):=sup{‖P‖Δd+1:P∈Pn(d+1),‖P‖K≤1},where Δ ^d+1 is the unit polydisk. For all x∈ [- 1 , 1] ^d with linearly independent entries, we have the lower estimate logEn(x)≥nd+1(d-1)!(d+1)logn-O(nd+1);for Diophantine x, we have logEn(x)≤nd+1(d-1)!(d+1)logn+O(nd+1).In particular, this estimate holds for almost all x with respect to Lebesgue measure. The results here generalize those of Coman and Poletsky [6] for d= 1 without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.