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
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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
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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
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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
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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
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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
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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
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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).
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A conjecture of Wilf on the Frobenius number
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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
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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
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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
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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