Year: 2025
Type: article
Abstract: Cilleruelo conjectured that for an irreducible polynomial f ∈ ℤ[ X ] of degree d ⩾ 2, denoting $${L_f}(N) = {\text{lcm}}({f(1),f(2), \ldots,f(N)})$$ L f ( N ) = lcm ( f ( 1 ) , f ( 2 ) , … , f ( N ) ) one has $$\log L_{f}(n)\sim(d-1)N\log N.$$ log L f ( n ) ∼ ( d − 1 ) N log N . He proved it in the case d = 2 but it remains open for every polynomial with d > 2. While the tight upper bound log L f ( n ) ≲ ( d − 1) N log N is known, the best known general lower bound due to Sah is $$\log L_{f}(n)\gtrsim N\log N.$$ log L f ( n ) ≳ N log N . We give an improved lower bound for a special class of irreducible polynomials, which includes the decomposable irreducible polynomials f = g ◦ h, g, h ∈ ℤ[ x ], deg g , deg h ≥ 2, for which we show $$\log {L_f}(n) \gtrsim {{d - 1} \over {d - \deg g}}N\log N.$$ log L f ( n ) ≳ d − 1 d − deg g N log N . We also improve Sah’s lower bound $$\log {\ell_f}(n) \gtrsim {{2} \over {d}} N\log N$$ log ℓ f ( n ) ≳ 2 d N log N for the radical ℓ f ( N ) = rad( L f ( N )) for all irreducible f with d
Source: Israel Journal of Mathematics
Author: Alexei Entin
Institution: Tel Aviv University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4416224151