### 2-D polynomial equations

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We study 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of $n$-gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus $0$ curves with $n$ marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.

It is proved that a Marot ring is a Krull ring if and only if its monoid of regular elements is a Krull monoid.

Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in $K[x{\u2081}^{p},...,x{\u2099}^{p}]$. We prove that $K[x{\u2081}^{p},...,x{\u2099}^{p},f]$ is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f are relatively prime. The proof is based on a generalization of Freudenburg’s lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.

Consider an experiment with d+1 possible outcomes, d of which occur with probabilities $x\u2081,...,{x}_{d}$. If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in $x\u2081,...,{x}_{d}$. We characterize those polynomials which arise as the probability of such an event. We use this to characterize those x⃗ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.

We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket $R$-module is $R$ tensor a bracket group.

We describe a cluster algebra algorithm for calculating $q$-characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra ${U}_{q}\left(\widehat{\U0001d524}\right)$. This yields a geometric $q$-character formula for tensor products of Kirillov–Reshetikhin modules. When $\U0001d524$ is of type $A,D,E$, this formula extends Nakajima’s formula for $q$-characters of standard modules in terms of homology of graded quiver varieties.

Let F=X-H:${k}^{n}$ → ${k}^{n}$ be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G1,...,Gn) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of ${G}_{i}$ of degree 2d+1 can be expressed as ${G}_{i}^{\left(d\right)}={\sum}_{T}\alpha {\left(T\right)}^{-1}{\sigma}_{i}\left(T\right)$, where T varies over rooted trees with d vertices, α(T)=CardAut(T) and ${\sigma}_{i}\left(T\right)$ is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, $F$ is an automorphism or, equivalently, ${G}_{i}^{\left(d\right)}$ is zero for sufficiently large d....

Let F = X + H be a cubic homogeneous polynomial automorphism from ${\u2102}^{n}$ to ${\u2102}^{n}$. Let $p$ be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that $deg{F}^{-1}\le {3}^{p-1}$. We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.

Using the notion of cyclically pure injective modules, a characterization of rings which are locally valuation rings is established. As applications, new characterizations of Prüfer domains and pure semisimple rings are provided. Namely, we show that a domain R is Prüfer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semisimple if and only if every R-module is cyclically pure...