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Let $k$ be an arbitrary field. In this note, we show that if a sequence of relatively prime positive integers ${bf a}=(a_1,a_2,a_3,a_4)$ defines a Gorenstein non complete intersection monomial curve ${mathcal C}({bf a})$ in ${mathbb A}_k^4$, then there exist two vectors ${bf u}$ and ${bf v}$ such that ${mathcal C}({bf a}+t{bf u})$ and ${mathcal C}({bf a}+t{bf v})$ are also Gorenstein non complete intersection affine monomial curves for almost all $tgeq 0$.
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We study a monomial derivation $d$ proposed by J. Moulin Ollagnier and A. Nowicki in the polynomial ring of four variables, and prove that $d$ has no Darboux polynomials if and only if $d$ has a trivial field of constants.
Let $mathcal{A}={{bf a}_1,ldots,{bf a}_n}subsetBbb{N}^m$. We give an algebraic characterization of the universal Markov basis of the toric ideal $I_{mathcal{A}}$. We show that the Markov complexity of $mathcal{A}={n_1,n_2,n_3}$ is equal to two if $I_{mathcal{A}}$ is complete intersection and equal to three otherwise, answering a question posed by Santos and Sturmfels. We prove that for any $rgeq 2$ there is a unique minimal Markov basis of $mathcal{A}^{(r)}$. Moreover, we prove that for any integer $l$ there exist integers $n_1,n_2,n_3$ such that the Graver complexity of $mathcal{A}$ is greater than $l$.
Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve $C$ in $mathbb{A}^3$ has Markov complexity $m(C)$ two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no $din mathbb{N}$ such that $m(C)leq d$ for all monomial curves $C$ in $mathbb{A}^4$. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in $mathbb{A}^n, ngeq 4$.
Let $k$ be a field and $G subseteq Gl_n(k)$ be a finite group with $|G|^{-1} in k$. Let $G$ act linearly on $A = k[X_1, ldots, X_n]$ and let $A^G$ be the ring of invariants. Suppose there does not exist any non-trivial one-dimensional representation of $G$ over $k$. Then we show that if $Q$ is a $G$-invariant homogeneous ideal of $A$ such that $A/Q$ is a Gorenstein ring then $A^G/Q^G$ is also a Gorenstein ring.
We show that a non-trivial fiber product $Stimes_k T$ of commutative noetherian local rings $S,T$ with a common residue field $k$ is Gorenstein if and only if it is a hypersurface of dimension 1. In this case, both $S$ and $T$ are regular rings of dimension 1. We also give some applications of this result.
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