On p -Frobenius of affine semigroups

<jats:p>This paper studies the <jats:italic>p</jats:italic>-Frobenius vector of affine semigroups <jats:inline-formula><jats:alternatives><jats:tex-math>$$S\subset \mathbb {N}^q$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>S</mml:mi>
                    <mml:mo>⊂</mml:mo>
                    <mml:msup>
                      <mml:mrow>
                        <mml:mi>N</mml:mi>
                      </mml:mrow>
                      <mml:mi>q</mml:mi>
                    </mml:msup>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>. Defined with respect to a graded monomial order, the <jats:italic>p</jats:italic>-Frobenius vector represents the maximum element with at most <jats:italic>p</jats:italic> factorizations within <jats:italic>S</jats:italic>. We develop efficient algorithms for computing these vectors and analyze their behavior under the gluing operations with <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {N}^q$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msup>
                    <mml:mrow>
                      <mml:mi>N</mml:mi>
                    </mml:mrow>
                    <mml:mi>q</mml:mi>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>


Definitions
If n ∈ S, then Z n (S) = {λ = (λ 1 , . . ., λ h ) The minimum integer cone containg S a affine semigroup is C(S) has always a finite number of extremal rays (there exist {τ 1 , . . ., τ r } ⊆ S generating C(S)) We fix ⪯ a monomial order on N q (a total order compatible with + in N q and such that 0 ⪯ x for all x ∈ N q ) 3/20 On p-Frobenius of affine semigroups Gluing semigroups

Frobenius elements and vectors
The Frobenius element of S a numerical semigroup is The Frobenius number f of S is the maximum integer f satisfying that On p-Frobenius of affine semigroups Gluing semigroups Bibliography

p-Frobenius vector
The p-Frobenius of S is the element (p ∈ N) (see [KY23] and [Bro+10]) Other definition: At least when A is not a m.s.g.{g p (S)} p∈N is not always an increasing sequence One of our goals is to provide algorithms for computing p-Frobenius vector in numerical semigroups and C-semigroups

Presentations of semigroups
Every finitely generated commutative monoid is isomorphic to a quotient of the form N h /σ with σ congruence on N h × N h , a equivalence relation compatible with the addition (see [RG99]) On p-Frobenius of affine semigroups Gluing semigroups

Presentations of semigroups
If we consider the S-graded polynomial ring, the S-homogeneus ideal This is used to know if a multiple of a generator can be expressed by using the other generators

Main result
Theorem Let S = ⟨a 1 , . . ., a h ⟩ ⊂ N q be an affine semigroup, p ∈ N \ {0}, and ⪯ a monomial ordering on N q .Then, there exists F p (S) if and only if for every k ∈ {1, . . ., h}, there exist ⇕ In every extremal ray there are at least two minimal generators of S On p-Frobenius of affine semigroups Gluing semigroups

Bibliography
Computation of F p (S) and g p (S) Input: A minimal system of generators {a 1 , . . ., a h } of S and p ∈ N \ {0}.Output: F p (S) and g p (S). G ← a generating set of the ideal Gluing semigroups

Optimizations
• D is the bounded set where we search for F p (S) and g p (S) Gluing semigroups

Bibliography
Improved computation of F 1 (S) and g 1 (S) Input: A minimal system of generators {a 1 , . . ., a h } of S.
Output: F 1 (S) and g 1 (S).if there is an extremal ray of C(S) with only one minimal generator of S then return ̸ ∃F 1 (S) and ̸ ∃g 1 (S) end if B ← a Gröbner basis of On p-Frobenius of affine semigroups Gluing semigroups

Bibliography
Indispensable binomials

Lemma ([OV10])
Let S such that there is an element m = h i=1 α i a i ∈ S with #Z h i=1 α i a i (S) = 2.Then, there is at least an indispensable binomial in I S .

Corollary
Given S an affine semigroup satisfying the hypothesis of Theorem 1.If there is no indispensable binomial in I S , then g 2 (S) = ∅ 12/20 On p-Frobenius of affine semigroups Gluing semigroups

Bibliography
Improved computation of g 2 (S) Input: A minimal system of generators {a 1 , . . ., a h } of S. Output: F 2 (S) and g 2 (S).G ← a generating set of the ideal Gluing semigroups

Bibliography
Improved computation of g 2 (S) We look for the elements of g 2 (S) in the set with Ω the set of exponents of the indispensable binomials and D the same set of the first algorithm p} and g p (S) = max ⪯ {n ∈ D | #Z n (S) = p} We sort the above set • Starting from the maximum and decreasing according to ⪯ we check if the number of expressions of the element is equal to p Improvements of this algorithm are done for p = 1 and p = 2 bX δ with b ∈ I then G ← G ∪ {{γ, γ ′ }} else D ← D \ {γ, γ ′ } end if end while return g 2 (S) = max ⪯ { h i=1 γ i a i | (γ 1 , . . ., γ h ) ∈ G and #Z h i=1 γ i a i(S) = 2} and F 2 (S) = max ⪯ {f , g 2 (S)}