For any $k\ge 1$, this paper studies the number of polynomials having k irreducible factors (counted with or without multiplicities) in ${\mathbf{F}}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for k in a certain range. In the generic case, the bias dissipates as the degree of the modulus or k gets large, but there are cases when the bias is extreme. In contrast to the case of products of k prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon.

中文翻译：

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Chebyshev 对不可约多项式乘积的偏见

对于任何 $克\ge 1$，本文研究具有多项式的数目ķ不可约因子（具有或不具有多重计数）在${\mathbf{F}}_q[吨]$在不同的算术级数之间。得到一定范围内k 的均匀计数函数差的渐近公式。在一般情况下，偏差会随着模数或k的程度变大而消散，但也有偏差非常大的情况。对比k个素数乘积的情况，我们在函数域设置中显示存在完全偏差，即差分函数可能具有常数符号。几个例子说明了这种新现象。