Novel bounds of algebraic Nikol'skii constant

Let $M_{n}=\sup_{P\in \mathcal{P}_{n}\setminus \{0\}} \frac{\max_{x\in [-1,1]}|P(x)|}{\int_{-1}^{1}|P(x)| dx}$ be the Nikol'skii constant between the uniform and integral norms for algebraic polynomials with complex coefficients of degree at most $n$. D. Amir and Z. Ziegler (1976) proved that $0.125(n+1)^{2}\le M_{n}\le 0.5(n+1)^{2}$ for $n\ge 0$. The same upper bound was obtained by T.K. Ho (1976). F. Dai, D. Gorbachev, and S. Tikhonov (2019–2020) refined this result by establishing that $M_{n}=Mn^{2}+o(n^{2})$ for $n\to \infty$, where $M\in (0.141,0.192)$ is the sharp Nikol'skii constant for entire functions of exponential spherical type in the space $L^{1}(\mathbb{R}^{2})$ and functions of exponential type in $L^{1}(\mathbb{R})$ with weight $|x|$.
We prove that for arbitrary $n\ge 0$ one has $M(n+1)^{2}\le M_{n}\le M(n+2)^{2}$, where $M\in (0.1410,0.1411)$. This statement also allows us to refine the exact Jackson–Nikol'skii constant for polynomials on the Euclidean sphere $\mathbb{S}^{2}$. The proof is based on the relationship between the algebraic Nikol'skii constants and the Bernstein–Nikol'skii trigonometric constants and our estimates of these constants (2018–2019). We also apply the characterization of the extremal algebraic polynomial obtained by D. Amir and Z. Ziegler (1976), V.V. Arestov and M.V. Deikalova (2015). Using this characterization, we compose a trigonometric system for determining the zeros of an extremal polynomial, which we solve approximately with the required accuracy using Newton's method.