Proving Simultaneous Positivity of Linear Forms

In this paper we study the computational effort involved in establishing that m linear forms ll(x),..., lm(x), where x ----(x 1 ,..., x,,), are simultaneously non-negative for a substitution y e R n, i.e. that 0 ~< li(y), 1 <~ i ~ m. We develop the notion of a complete proof for simultaneous positivity and show that under certain natural conditions, proofs for the simultaneous positivity ofm forms will always requirem steps. Appropriate theorems and counterexamples show that if the conditions in the main theorem do not hold, then the conclusion need not hold. A number of applications of the main theorem are given. The present discussion brings out ideas and concepts which may be of interest and applicability beyond the present context. We shall devote some time to present these ideas here. Let P be a computational problem. We can distinguish between three levels of treatment of the problem. The highest level is finding a solution S to the problem. Depending on P, the solution S may be a number, a permutation, a truth value, etc. I f the algorithm for arriving at the solution S is a valid one, then the fact that S was produced, is in itself also a proof that S is indeed a solution of the problem P. The next level is that of checking a proposed solution. Given a possible solution S for the problem P, we wish to check whether S is indeed a solution. As an illustration of the difference between finding a solution and checking a proposed solution, consider the problem of finding a proper divisor of an integer n known to be composite. The ordinary algorithm for finding a divisor of n may take up to about ~/n steps. However, if given a supposed divisor k of n, then to check it we just have to divide n by k.