We consider systems of polynomial equations and inequalities to be solved in integers. By applying the circle method, when the number of variables is large and the system is geometrically well-behaved we give an asymptotic estimate for the number of solutions of bounded size. In the case of R homogeneous equations having the same degree d, a classic theorem of Birch provides such an estimate provided the number of variables is R(R+1)(d-1)2d-1+R or greater and the system is nonsingular. In many cases this conclusion has been improved, but except in the case of diagonal equations the number of variables needed has always grown quadratically in R. We give a result requiring only d2dR+R variables, obtaining linear growth in R. When d = 2 or 3 we require only that the system be nonsingular; when d&LT;4 we require that the coefficients of the equations belong to a certain explicit Zariski open set. These conditions are satisfied for typical systems of equations, and can in principle be checked algorithmically for any particular system. We also give an asymptotic estimate for the number of solutions to R polynomial inequalities of degree d with real coefficients, in the same number of variables and satisfying the same geometric conditions as in our work on equations. Previously one needed the number of variables to grow super-exponentially in the degree d in order to show that a nontrivial solution exists.</p
