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We investigate the complexity of integration and derivative for multivariatepolynomials in the standard computation model. The integration is in the unitcube $[0,1]^d$ for a multivariate polynomial, which has format $f(x_1,\cdots,x_d)=p_1(x_1,\cdots, x_d)p_2(x_1,\cdots, x_d)\cdots p_k(x_1,\cdots, x_d)$,where each $p_i(x_1,\cdots, x_d)=\sum_{j=1}^d q_j(x_j)$ with all singlevariable polynomials $q_j(x_j)$ of degree at most two and constantcoefficients. We show that there is no any factor polynomial time approximationfor the integration $\int_{[0,1]^d}f(x_1,\cdots,x_d)d_{x_1}\cdots d_{x_d}$unless $P=NP$. For the complexity of multivariate derivative, we consider thefunctions with the format $f(x_1,\cdots, x_d)=p_1(x_1,\cdots,x_d)p_2(x_1,\cdots, x_d)\cdots p_k(x_1,\cdots, x_d),$ where each$p_i(x_1,\cdots, x_d)$ is of degree at most $2$ and $0,1$ coefficients. We alsoshow that unless $P=NP$, there is no any factor polynomial time approximationto its derivative ${\partial f^{(d)}(x_1,\cdots, x_d)\over \partial x_1\cdots\partial x_d}$ at the origin point $(x_1,\cdots, x_d)=(0,\cdots,0)$. Ourresults show that the derivative may not be easier than the integration in highdimension. We also give some tractable cases of high dimension integration andderivative.

Multivariate Polynomial Integration and Derivative Are Polynomial Time Inapproximable unless P=NP