Tangential Markov inequalities on transcendental curves (Q1404478)

Let \(M\) be a smooth manifold in \(\mathbb R^d\). We say that \(M\) admits a tangential Markov inequality of exponent \(\ell\) if there is a constant \(C> 0\) such that, for all polynomials \(P\in \mathbb R[x_1,\dots, x_d]\) and points \(a\in M\), \(| D_T P(a)|\leq C(\deg P)^\ell\| P\|_M\). Here \(D_TP\) denotes any (unit) tangential derivative of \(P\) and \(\| P\|_M\) is the supremum norm of \(P\) on \(M\). It is known [see \textit{L. Bos}, \textit{N. Levenberg}, \textit{P. Milman} and \textit{B. A. Taylor}, Indiana Univ. Math. J. 44, No. 1, 115--138 (1995; Zbl 0824.41015), \textit{M. Baran} and \textit{W. Pleśniak}, Stud. Math. 141, No. 3, 221--234 (2000; Zbl 0987.41005)] that such tangential Markov inequalities of exponent one characterize \(M\) being algebraic. On the other hand, not all analytic manifolds admit a tangential Markov inequality (of any exponent \(\ell\)) (see the above-mentioned paper by L. Bos, N. Nevenberg, P. Milman, B. A. Taylor). Up to now it was not clear whether or not the existence of a Markov inequality of some exponent mean that \(M\) was (possibly singular) algebraic. The main result of this paper shows that this is not the case. Namely, the authors prove that on the curves \(\gamma:= (x, e^{t(x)})\), \(x\in [a,b]\), where \(t(x)\) is a fixed polynomial, there holds a tangential Markov inequality of exponent four for algebraic polynomials \(P_N(x,y)\) of degree of at most \(N\) in each variable \(x,y:\|(P_N(x, e^{t(x)}))'\|_{[a,b]}\leq CN^4\| P_N\|_\gamma\), and the exponent four is sharp. On the other hand, they show that the corresponding tangential Markov factors on curves \(y= x^\alpha\) with irrational \(\alpha\) grow exponentially in the degree of the polynomials. Reviewer's remark: One can also show that the sharpness of the exponent four above follows from a general property of Markov-type exponents based on a Krasnoselski-Krein lemma and a uniform version of the Bernstein-Walsh-Siciak theorem about polynomial approximation of analytic functions [see the reviewer, An estimate from below of a generalized Markov exponent, Institutive of Mathematics, Jagiellonian University, Cracow, preprint 2002].