Square-free values of polynomials had been studied by various authors, including Estermann, Heath-Brown and Hooley. For $1\le x,y\le H$, Tolev proved that the number of the square-free values attained by the polynomial $x^2+y^2+1$ has the asymptotic formula $c_1{H}^2+O(H^{4/3+\epsilon })$, where is $c_1$ is an absolute constant and ε is an arbitrary small positive number. The key ingredient of his proof which leads to the elaborate error term is the estimate for the Kloosterman sum. In this paper, by using Tolev's method and some estimate for the Salié sum, we show that for any fixed integer k, there is an absolute constant $c_2$ such that the number of square-free values of the polynomial $x^2+y^2+z^2+k$ with $1\le x,y,z\le H$ is $c_2{H}^3+O(H^{7/3+\epsilon })$.

多项式的无平方值已被包括 Estermann、Heath-Brown 和 Hooley 在内的多位作者研究过。为了$1\le X,\mathrm{是的}\le H$, Tolev 证明了多项式获得的无平方值的数量$X^2+{\mathrm{是的}}^2+1$有渐近公式$C_1{H}^2+○(H^{4/3+\epsilon })$， 哪里$C_1$是一个绝对常数，ε是一个任意小的正数。他的证明中导致复杂误差项的关键因素是对 Kloosterman 和的估计。在本文中，通过使用 Tolev 方法和对 Salié 和的一些估计，我们证明对于任何固定整数k，都有一个绝对常数$C_2$使得多项式的无平方值的数量$X^2+{\mathrm{是的}}^2+z^2+ķ$和$1\le X,\mathrm{是的},z\le H$是$C_2{H}^3+○(H^{7/3+\epsilon })$.