Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k \geq 2$ and suppose that there is no prime $p$ such that $p^k$ divides $F(a,b)$ for all pairs of integers $(a,b)$. Let $R_{F,k}(Z)$ denote the number of $k$-free integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_{F,k}$ such that $R_{F,k}(Z)$ is asymptotic to $C_{F,k} Z^{2/d}$ provided that $k$ exceeds ${7r}/{18}$ or $(k,r)$ is $(2,6)$ or $(3,8)$.

中文翻译：

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关于无$ k $整数的二进制形式表示

假设$ F $是具有整数系数，非零判别力和度d $且其中$ d $至少为$ 3 $的二进制形式，并且$ r $表示$ F $不可推论因子相对于有理数的最大程度。假设$ k $是一个带有$ k \ geq 2 $的整数，并假设没有素数$ p $使得对于所有整数对$（a，b，$ p ^ k $都将​​$ F（a，b）$除以$ F（a，b）$ ）$。令$ R_ {F，k}（Z）$表示最多$ Z $的无$ k $绝对值的整数，其整数由$ F $表示。我们证明存在一个正数$ C_ {F，k} $使得$ R_ {F，k}（Z）$渐近于$ C_ {F，k} Z ^ {2 / d} $ k $超过$ {7r} / {18} $或$（k，r）$是$（2,6）$或$（3,8）$。