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Register a new userHasse-Schmidt Derivations and Coefficient Fields in Positive Characteristics
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We show how to express any Hasse-Schmidt derivation of an algebra in terms of a finite number of them under natural hypothesis. As an application, we obtain coefficient fields of the completion of a regular local ring of positive characteristic in terms of Hasse-Schmidt derivations
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Let k be a perfect field of positive characteristic, k(t)_{per} the perfect closure of k(t) and A=k[[X_1,...,X_n]]. We show that for any maximal ideal N of A=k(t)_{per}otimes_k A, the elements in hat{A_N} which are annihilated by the Taylor Hasse-Schmidt derivations with respect to the X_i form a coefficient field of hat{A_N}.
Let K be a field of characteristic zero. We prove that images of a linear K-derivation and a linear K-E-derivation of the ring K[x 1 ,x 2 ,x 3 ] of polynomial in three variables over K are Mathieu-Zhao subspaces, which affirms the LFED conjecture for linear K-derivations and linear K-E-derivations of K[x 1 ,x 2 ,x 3 ].
Affine ind-varieties are infinite dimensional generalizations of algebraic varieties which appear naturally in many different contexts, in particular in the study of automorphism groups of affine spaces. In this article we introduce and develop the basic algebraic theory of topologically integrable derivations of complete topological rings. We establish a bijective algebro-geometric correspondence between additive group actions on affine ind-varieties and topologically integrable derivations of their coordinate pro-rings which extends the classical fruitful correspondence between additive group actions on affine varieties and locally nilpotent derivations of their coordinate rings.
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 $L$ be a finite extension of $mathbb{F}_q(t)$. We calculate the proportion of polynomials of degree $d$ in $mathbb{F}_q[t]$ that are everywhere locally norms from $L/mathbb{F}_q(t)$ which fail to be global norms from $L/mathbb{F}_q(t)$.
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