Let \(h_1(y),\ldots,h_s(y)\) be polynomials with real coefficients, and put \(H({\mathbf y})=H(y_1,\ldots,y_s)=h_1(y_1)+\cdots+h_s(y_s)\). Suppose throughout that the degree of each \(h_i(y)\) is at most \(k\) and at least one, and that there exists a couple of coefficients of non-constant terms of \(H({\mathbf y})\) such that the ratio of them is irrational. If all the \(h_i\) are of even degree and all the leading coefficients of the \(h_i\) are of the same sign, then \(H({\mathbf y})\) is called a positive- or negative-definite polynomial, according to the sign of the leading coefficients of the \(h_i\). Otherwise, \(H({\mathbf y})\) is called an indefinite polynomial. Now let \(\varepsilon\) be any given positive number, and let \(P\) denote any positive number which is sufficiently large in terms of \(k\), \(\varepsilon\) and the coefficients of \(H({\mathbf y})\). Then it is proved in this paper that there exists a positive integer \(s_0(k)\), depending only on \(k\), such that whenever \(s\geq s_0(k)\), the following two conclusions hold for certain positive numbers \(C_1\), \(C_2\) and \(C_3\) with \(C_2