R-fat Linearized Polynomials over Finite Fields
R-fat Linearized Polynomials over Finite Fields
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DOI：
10.1016/j.jcta.2022.105609
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发表时间：
2020-12
期刊：
影响因子：
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通讯作者：
D. Bartoli;Giacomo Micheli;Giovanni Zini;Ferdinando Zullo
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文献类型：
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作者：
D. Bartoli;Giacomo Micheli;Giovanni Zini;Ferdinando Zullo
摘要r-胖多项式是离散多项式的自然推广。他们定义线性集的投影线PG（1，q n）的秩n与r点的重量大于1。利用代数曲线和函数域上的技巧，我们得到了r的数值界和r> 0时例外r-fat多项式的不存在性。当考虑F q 4上的线性化多项式时，我们完全确定了r的可能值，并且我们还在PG（1，q 5）中提供了一族1-fat多项式。此外，我们研究了LP-多项式（即f（x）= x+ δ x q 2 s∈ F q n [x]，gcd n（n，s）= 1的多项式），确定了这样的多项式是r-fat的值r的谱。
Abstract r-fat polynomials are a natural generalization of scattered polynomials. They define linear sets of the projective line PG (1, q n) of rank n with r points of weight larger than one. Using techniques on algebraic curves and function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with r> 0. We completely determine the possible values of r when considering linearized polynomials over F q 4 and we also provide one family of 1-fat polynomials in PG (1, q 5). Furthermore, we investigate LP-polynomials (ie polynomials of type f (x)= x+ δ x q 2 s∈ F q n [x], gcd(n, s)= 1), determining the spectrum of values r for which such polynomials are r-fat.