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تسجيل مستخدم جديدOn the enumeration of polynomials with prescribed factorization pattern
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We use generating functions over group rings to count polynomials over finite fields with the first few coefficients prescribed and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic $n$-smooth polynomial of degree $m$ over a finite field, as well as the number of monic $n$-smooth polynomial of degree $m$ with the prescribed trace coefficient.
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We obtain estimates on the number $|mathcal{A}_{boldsymbol{lambda}}|$ of elements on a linear family $mathcal{A}$ of monic polynomials of $mathbb{F}_q[T]$ of degree $n$ having factorization pattern $boldsymbol{lambda}:=1^{lambda_1}2^{lambda_2}cdots n
^{lambda_n}$. We show that $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}/{2}})$, where $mathcal{T}(boldsymbol{lambda})$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $boldsymbol{lambda}$ and $m$ is the codimension of $mathcal{A}$. Furthermore, if the family $mathcal{A}$ under consideration is sparse, then $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}})$. Our estimates hold for fields $mathbb{F}_q$ of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the $mathcal{O}$--notation in terms of $boldsymbol{lambda}$ and $mathcal{A}$ with good behavior. Our approach reduces the question to estimate the number of $mathbb{F}_q$--rational points of certain families of complete intersections defined over $mathbb{F}_q$. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of $mathbb{F}_q$--rational points are established.
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