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A conjecture of Wilf on the Frobenius number Alessio Sammartano Dalhousie University - October 19, 2014 AMS Fall Eastern Sectional Meeting Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 1 / 10 Based on: A. Sammartano, Numerical semiroups with large embedding dimension satisfy Wilf’s conjecture, Semigroup Forum 85, 439–447 (2012). A. Moscariello, A. Sammartano, On a conjecture by Wilf about the Frobenius number, submitted, arXiv:1408.5331. Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 2 / 10 The Diophantine Frobenius problem Let a1 , a2 , . . . , ad be coprime positive integers. The Frobenius number is ( d )! X F = max Z \ λi ai : λi ∈ N i=1 Example: F (a1 , a2 ) = a1 a2 − a1 − a2 (Sylvester, 1884). Problem: find bounds, asymptotics, algorithms for F . Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 3 / 10 The Diophantine Frobenius problem Let a1 , a2 , . . . , ad be coprime positive integers. The Frobenius number is ( d )! X F = max Z \ λi ai : λi ∈ N i=1 Example: F (a1 , a2 ) = a1 a2 − a1 − a2 (Sylvester, 1884). Problem: find bounds, asymptotics, algorithms for F . Let n = Card [0, F ] ∩ ( d X )! λi ai : λi ∈ N i=1 Herbert S. Wilf (1931-2012) proposed the following: Conjecture (Wilf, 1978) The following inequality holds: F + 1 ≤ nd. Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 3 / 10 Example Let {a1 , a2 , a3 , a4 } = {6, 8, 9, 19}. 0 6 8 9 12 13 14 → · · · Thus d = 4, n = 5, F = 13 and F + 1 ≤ nd. Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 4 / 10 A length inequality Consider the local ring of a monomial curve R = k[[t a1 , t a2 , . . . , t ad ]] Q = k((t)), R = k[[t]] The conductor of R in R is C = (R :Q R) and `(R/C) ≤ `(R/R). Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 5 / 10 A length inequality Consider the local ring of a monomial curve R = k[[t a1 , t a2 , . . . , t ad ]] Q = k((t)), R = k[[t]] The conductor of R in R is C = (R :Q R) and `(R/C) ≤ `(R/R). Wilf’s conjecture is equivalent to the following: Conjecture (Wilf, 1978) The following inequality holds: `(R/R) ≤ (edim(R) − 1)`(R/C). Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 5 / 10 Known cases Denote by e(R) the multiplicity of R. The conjecture holds in the following cases: edim(R) = 2 (Sylvester, 1884); edim(R) = 3 (Fröberg, Gottlieb and Häggkvist, 1986); edim(R) = e(R) (Dobbs and Matthews, 2003); 2`(R/R) < 3e(R) or `(R/C) ≤ 2e(R) (Kaplan, 2012); `(R/C) ≤ 10 (Eliahou, 2014). Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 6 / 10 Main Results (1) Theorem (S.) If e(R) ≤ 2edim(R) then the inequality `(R/R) ≤ (edim(R) − 1)`(R/C) holds. Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 7 / 10 Main Results (1) Theorem (S.) If e(R) ≤ 2edim(R) then the inequality `(R/R) ≤ (edim(R) − 1)`(R/C) holds. Corollary If e(R) is large enough with respect to `(R/C) then the inequality holds. Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 7 / 10 Main Results (2) Theorem (Moscariello, S.) Let ρ = l e(R) edim(R) m . The following inequality ` R/R ≤ (edim(R) − 1)`(R/C) holds if e(R) is large enough with respect to ρ and e(R) is not divisible by a certain finite set of primes. Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 8 / 10 Equality in the conjecture Question Does the equality ` R/R = (edim(R) − 1)`(R/C) hold if and only if one of the following conditions occurs: edim(R) ≤ 2; there exists x ∈ m and p ∈ N such that m = (x) + C and CR = x p R. Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 9 / 10 Equality in the conjecture Question Does the equality ` R/R = (edim(R) − 1)`(R/C) hold if and only if one of the following conditions occurs: edim(R) ≤ 2; there exists x ∈ m and p ∈ N such that m = (x) + C and CR = x p R. The question has an affirmative answer in the following cases: ` R/R ≤ 35; under the assumptions of the two theorems. Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 9 / 10 A related inequality Theorem (Matsuoka; Fröberg et al.; Brown-Herzog) If R be a 1-dimensional Cohen-Macaulay local ring with type t(R) and such that R is a finite R-module, then ` R/R ≤ t(R)`(R/C). Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 10 / 10 A related inequality Theorem (Matsuoka; Fröberg et al.; Brown-Herzog) If R be a 1-dimensional Cohen-Macaulay local ring with type t(R) and such that R is a finite R-module, then ` R/R ≤ t(R)`(R/C). Question Let R be a 1-dimensional Cohen-Macaulay local ring such that R is a finite R-module. Under what assumptions do we have ` R/R ≤ (edim(R) − 1)`(R/C)? Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 10 / 10 A related inequality Theorem (Matsuoka; Fröberg et al.; Brown-Herzog) If R be a 1-dimensional Cohen-Macaulay local ring with type t(R) and such that R is a finite R-module, then ` R/R ≤ t(R)`(R/C). Question Let R be a 1-dimensional Cohen-Macaulay local ring such that R is a finite R-module. Under what assumptions do we have ` R/R ≤ (edim(R) − 1)`(R/C)? Thank you! Alessio Sammartano (Purdue University) A conjecture of Wilf on the Frobenius number 10 / 10