Given a multivariate rational function $p(\overrightarrow{x})=\frac{f(\overrightarrow{x})}{g(\overrightarrow{x})}$ over $[0,1{]}^{n}$ with $p(\overrightarrow{x})\in [0,1]$, how can we come up with a polynomial approximation of $p$, say $q(\overrightarrow{x})$ such that $|p(\overrightarrow{x})-q(\overrightarrow{x})|\le \u03f5$ for all $\overrightarrow{x}$ ?

For the univariate case, we might use chebyshev approximation, however, what are the results for the multivariate case?

For the univariate case, we might use chebyshev approximation, however, what are the results for the multivariate case?