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Let $3leq d_1leq d_2leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $frac{d_1}{gcd(d_1,d_3)} eq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3in d_1mathbb{N}+d_2mathbb{N}$; (2) If $d_3$ is a prime number and $gcd(d_1,d_2)=1$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_3in d_1mathbb{N}+d_2mathbb{N}$. We also relate this investigation with a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example to this conjecture.
Let $(a,a+d,a+2d)$ be an arithmetic progression of positive integers. The following statements are proved: (1) If $amid 2d$, then $(a, a+d, a+2d)inmdeg(Tame(mathbb{C}^3))$. (2) If $a mid 2d$, then, except for arithmetic progressions of the form $
In this paper we study principally polarized abelian varieties that admit an automorphism of prime order $p>2$. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that
This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of $H(K)$, any dg
We present a library autgradalg.lib for the free computer algebra system Singular to compute automorphisms of integral, finitely generated $mathbb{C}$-algebras that are graded pointedly by a finitely generated abelian group. It implements the algorit
The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $mathbb Q[x_1, dots, x_n]$ to a corresponding ideal in $mathbb F_p[x_1,dots, x_n]$ where $p$ is a prime number; in other words, the textit{reduction modulo