SMALL ZEROS OF ADDITIVE FORMS IN SEVERAL VARIABLES

Let f(X) be an additive form defined by f(X) = f(x(1), x(2),...,x(s)) = sigma(1)a(1)x(1)(k) + sigma(2)a(2)x(2)(k) +...+ a(s)a(s)x(s)(k), whe re a(i) not equal 0 is integer, i = 1, 2,...,s. In 1979, Schmidt prove d that if epsilon > 0 then there is a large constant C(k, epsilon) suc h that for s greater than or equal to C(k, epsilon) the equation f(X) = 0 has a nontrivial integer solution in sigma(1), sigma(2),...,sigma( s), x(1), x(2),...,x(s) satisfying sigma(i) = +/-1 and \x(i)\ less tha n or equal to A(epsilon), i=1, 2...,s where A = max/1 less than or equ al to i less than or equal to s \a(i)\. Schmidt did not estimate this constant C(k, epsilon) since it would be extremely large. In this pape r, we prove the following result: Theorem 1 If logA less than or equal to 1/epsilon, then C(k, epsilon) less than or equal to max (2/epsilon , 20); if log A > 1/epsilon--A is sufficiently large and 1/2 A less th an or equal to \a<INF>i</INF>\ less than or equal to A,i = 1, 2,..., s , then C(k, epsilon) less than or equal to c<INF>1</INF>c<INF>2</INF>< SUP>p</SUP>; and if the last condition is omitted, then C(k, epsilon) less than or equal to [4/epsilon] c<INF>1</INF>c<INF>2</INF><SUP>p</SU P>,<SUP></SUP> where [GRAPHICS] c(2) = 100c(1)k(2).2(k) + c(1)(2) and p = 2 [log 2c(1)/epsilon]. We note that the inequality in the first ca se is sharp but not in the second case.