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We consider the nonlinear integer programming problem of minimizing aquadratic function over the integer points in variable dimension satisfying asystem of linear inequalities. We show that when the Graver basis of the matrixdefining the system is given, and the quadratic function lies in a suitable{\em dual Graver cone}, the problem can be solved in polynomial time. Wediscuss the relation between this cone and the cone of positive semidefinitematrices, and show that none contains the other. So we can minimize inpolynomial time some non-convex and some (including all separable) convexquadrics.  We conclude by extending our results to efficient integer minimization ofmultivariate polynomial functions of arbitrary degree lying in suitable cones.

The quadratic Graver cone, quadratic integer minimization, and extensions