Let Bbbk have characteristic zero, let R=Bbbk[x,y,z], and let I be generated by x^(a₁+t), y^(a₂+t), z^(a₃+t) and x^(a₁)y^(a₂)z^(a₃), so that R/I is a level monomial almost complete intersection. Migliore, Miró-Roig and Nagel conjectured a characterisation o…
Commutative AlgebraAI reviewed · Human review open
Let Bbbk have characteristic zero, let E=Bbbk[x₁,…,xₘ]/(x₁²,…,xₘ²), let e=x₁+…+xₘ, and let A=E/(e²), so that Aⱼ=Eⱼ/e²Eⱼ₋₂. Crispin Quiñonez, Lundqvist and Nenashev conjectured that the Hilbert function of Bbbk[x₁,…,xₙ]/(x₁²,…,xₙ²,ℓ₁²,ℓ₂²), for general linea…
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Let R be an integral domain and f: Rsetminus{0} → ℤ_(≥ 0) a Euclidean function in the modern sense, so that submultiplicativity f(a) ≤ f(ab) is not assumed. Following Sekhon, call f ultra-Euclidean if f(a+b) ≤ max{f(a),f(b)} whenever a,b,a+b ≠ 0, and write…
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Let H be a numerical semigroup, K[H] its semigroup ring and grₘ K[H] its tangent cone; H is called quadratic when the defining ideal I^(*)_(H) of grₘ K[H] is generated by quadrics. Stamate gave a maximal list of twelve h-vectors for a quadratic H of embeddi…
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Let n(g,t) be the number of numerical semigroups of genus g and type t. Chappelon, Ramírez Alfonsín and Stamate proved that each diagonal t=g-ℓ of this array is eventually constant, n(g,g-ℓ)=|V(ℓ)| for g ≥ 3ℓ-1, where V(ℓ) is a set of balanced binary words…
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Ripke and Yoon recently determined that exactly 134 distinct graded Betti tables occur among the 208 nonzero proper squarefree monomial ideals in five variables up to relabelling of the variables, and closed their paper with several questions, two of which…
Commutative AlgebraAI reviewed · Human review open
