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Register a new userA Note on Darboux Polynomials of Monomial Derivations
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Jiantao Li
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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.
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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 $tgeq 0$.
We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $Isubset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $Isubset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.
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