A characterization of linearized polynomials with maximum kernel
A characterization of linearized polynomials with maximum kernel
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DOI：
10.1016/j.ffa.2018.11.009
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发表时间：
2018-06
期刊：
影响因子：
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通讯作者：
Bence Csajbók;G. Marino;O. Polverino;Ferdinando Zullo
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文献类型：
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作者：
Bence Csajbók;G. Marino;O. Polverino;Ferdinando Zullo
我们给出了一个q多项式f / fqn的系数的充要条件，保证fqn中f的不同根的个数等于f的阶。我们说这些多项式有最大核。作为应用，我们详细研究了qn−2 / fqn阶的最大核多项式，对于n≤6，我们列出了所有具有最大核的q多项式。我们也得到了任意q-多项式的分裂域的信息。对于q - s多项式也证明了类似的结果，其中gcd (s, n)= 1。
We provide sufficient and necessary conditions for the coefficients of a q-polynomial f over F q n which ensure that the number of distinct roots of f in F q n equals the degree of f. We say that these polynomials have maximum kernel. As an application we study in detail q-polynomials of degree q n− 2 over F q n which have maximum kernel and for n≤ 6 we list all q-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary q-polynomial. Analogous results are proved for q s-polynomials as well, where gcd(s, n)= 1.