On the enumeration of polynomials with prescribed factorization pattern

Kuttner, Simon; Wang, Qiang

doi:10.1016/j.ffa.2022.102030

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2025

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Cited by 2 publications

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“…Formulas in terms of generating functions for the number of degree n polynomials over a finite field with prescribed coefficients and more general factorization patterns can be found in [6]. Related results on irreducible polynomials over finite fields can be found in [4].…”

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confidence: 99%

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Applications Of The Subset Sum Problem Over Finite Abelian Groups

Kuttner¹

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Given a finite abelian group G, a finite set D, and a mapping f : D → G, we find the number of r-subsets S ⊆ D where for b ∈ G,We obtain simple exact expressions when f is an abelian group homomorphism. When G = F q , we extend known results when D ∈ {F q , F * q } and f (x) = x N , which include quadratic and semiprimitive cases. We count degree n monic polynomials over F q with r distinct roots in a set D ⊆ F q when the leading terms of degree at least n − are fixed.We obtain new formulas for = 1 when D is a multiplicative subgroup of F * q , and for = 2 when D is an arbitrary subfield of F q with q odd.

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confidence: 99%

“…when D is an arbitrary subfield of F q and γ 1 , γ 2 ∈ F q (see Theorems 4.7 and 4.9). We obtain new expressions V r,D (γ 1 , γ 2 ) and W r,D (γ 1 , γ 2 ) as defined in Proposition 1.3 using the number of solutions to Equation (6). Additionally, our formula for V r,Fq (0, 0) in Theorem 4.5 is simpler than Theorem 2.11 in [15].…”

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confidence: 99%