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

We investigate the complexity of integration and derivative for multivariate polynomials in the standard computation model. The integration is in the unit cube [0, 1] for a multivariate polynomial, which has format f(x1, · · · , xd) = p1(x1, · · · , xd)p2(x1, · · · , xd) · · · pk(x1, · · · , xd), where each pi(x1, · · · , xd) = ∑d j=1 qj(xj) with all single variable polynomials qj(xj) of degree at most two and constant coefficients. We show that there is no any factor polynomial time approximation for the integration ∫ [0,1]d f(x1, · · · , xd)dx1 · · · dxd unless P = NP. For the complexity of multivariate derivative, we consider the functions with the format f(x1, · · · , xd) = p1(x1, · · · , xd)p2(x1, · · · , xd) · · · pk(x1, · · · , xd), where each pi(x1, · · · , xd) is of degree at most 2 and 0, 1 coefficients. We also show that unless P = NP, there is no any factor polynomial time approximation to its derivative ∂f (x1,···,xd) ∂x1···∂xd at the origin point (x1, · · · , xd) = (0, · · · , 0). Our #P -hard result for derivative shows that the derivative is not be easier than the integration in high dimension. We also give some tractable cases of high dimension integration and derivative.