Abstract
LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ n(λ) i=1 A k iλ μ whereA iλ are distinct primes. Let ϱλ(A) =n(λ) − 1 and let ρ(A) = Σλɛσ(A)ϱλ(A). The main result is the following:
Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.
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Stein, Y. The total reducibility order of a polynomial in two variables. Israel J. Math. 68, 109–122 (1989). https://doi.org/10.1007/BF02764973
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DOI: https://doi.org/10.1007/BF02764973