相关论文: Semifields and a theorem of Abhyankar

This paper has two objectives: we first generalize the theory of Abhyankar-Moh to quasi-ordinary polynomials, then we use the notion of approximate roots and that of generalized Newton polygons in order to prove the embedding conjecture for…

We study Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. For zero dimensional valuations we prove that whenever the ring is complete and the semigroup of values taken by the…

For an arbitrary hypersurface singularity, we construct a family of semigroups associated with algebraically closed fields that arise as an infinite union of rings of series. These semigroups extend the value semigroup of a plane curve…

The work proves that, for three-dimensional upper triangular groups over a field of odd characteristic with an abelian unipotent subgroup, the ring of invariants is polynomial if and only if the unipotent subgroup is generated by…

In this paper we introduce a model theoretic construction for the theories of uniform layered domains and semifields introduced in the paper of Izhakian, Knebusch and Rowen. We prove that, for a given layering semiring L, the theory of…

We show that the semi-simplicity conjecture for finitely generated fields follows from the conjunction of the semi-simplicity conjecture for finite fields and for the maximal abelian extension of the field of rational numbers.

We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively…

We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with…

A quasi-order is a binary, reflexive and transitive relation. In the Journal of Pure and Applied Algebra 45 (1987), S.M. Fakhruddin introduced the notion of (totally) quasi-ordered fields and showed that each such field is either an ordered…

This paper provides answers to several open problems about equational theories of idempotent semifields. In particular, it is proved that (i) no equational theory of a non-trivial class of idempotent semifields has a finite basis; (ii)…

The {\em superspace ring} of rank $n$ is the algebra $\Omega_n$ of differential forms on affine $n$-space. The algebra $\Omega_n$ is bigraded with respect to polynomial and exterior degree and carries a natural action of the symmetric group…

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and $\mathbb Q[X]$ with the property that its residue fields of prime characteristic are finite fields is equal to a generalized ring of integer-valued…

We previously showed that the inverse limit of standard-graded polynomial rings with perfect coefficient field is a polynomial ring, in an uncountable number of variables. In this paper, we show that the same result holds with arbitrary…

We show that the tensor product of modules of tensor fields is a noetherian module as a module over any graded Lie subalgebra of finite codimension in the Lie algebra of polynomial vector fields on $\mathbb{R}^n$. As a corollary, we prove…

We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can…

The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The…

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition…

We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we determine which hyperfields of order at most 4 are quotients…

To what extent does the maximal subfield spectrum of a division algebra determine the isomorphism class of that algebra? It has been shown that over some fields a quaternion division algebra's isomorphism class is largely if not entirely…

We introduce a ring and a field, generated by a semigroup, and we investigate some of their properties.