Introduction A system of homogeneous polynomials with rational coefficients has a non-trivial rational zero provided only that these polynomials are of odd degree, and the system has sufficiently many variables in terms of the number and degrees of these polynomials. While this striking theorem of Birch [1] addresses a fundamental diophantine problem in engagingly simple fashion, the problem of determining a satisfactory bound for the number of variables which suffice to guarantee the existence of a non-trivial zero remains unanswered in any but the simplest cases. Sophisticated versions of the Hardy–Littlewood method have been developed, first by Davenport [4] to show that 16 variables suffice for a single cubic form, and more recently by Schmidt [10] to show that (10r)5 variables suffice for a system of r cubic forms. Unfortunately even Schmidt's highly developed version of the Hardy-Littlewood method is discouragingly ineffective in handling systems of higher degree (see [11, 12]). The object of this paper is to provide a method for obtaining explicit bounds for the number of variables required in Birch's Theorem. Our approach to this problem will involve the Hardy–Littlewood method only indirectly, being motivated by the elementary diagonalisation method of Birch. Although it has always been supposed that Birch's method would necessarily lead to bounds too large to be reasonably expressed, we are able to reconfigure the method so as to obtain estimates which in general are considerably sharper than those following from Schmidt's methods (see forthcoming work [15] for amplification of this remark). Indeed, for systems of quintic forms our new bounds might, at a stretch, be considered “reasonable”.