Shallow neural network representation of polynomials

We show that $d$-variate polynomials of degree $R$ can be represented on $[0,1]^d$ as shallow neural networks of width $d+1+\sum_{r=2}^R\binom{r+d-1}{d-1}[\binom{r+d-1}{d-1}+1]$. Also, by SNN representation of localized Taylor polynomials of univariate $C^\beta$-smooth functions, we derive for shallow networks the minimax optimal rate of convergence, up to a logarithmic factor, to unknown univariate regression function.