### 2-D polynomial equations

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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.

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.

We give a simple geometric proof of Mohan Kumar's result about complete intersections.

Let $K$ be a field, $A=K[{X}_{1},\cdots ,{X}_{n}]$ and $\mathbb{M}$ the set of monomials of $A$. It is well known that the set of monomial ideals of $A$ is in a bijective correspondence with the set of all subsemiflows of the $\mathbb{M}$-semiflow $\mathbb{M}$. We generalize this to the case of term ideals of $A=R[{X}_{1},\cdots ,{X}_{n}]$, where $R$ is a commutative Noetherian ring. A term ideal of $A$ is an ideal of $A$ generated by a family of terms $c{X}_{1}^{{\mu}_{1}}\cdots {X}_{n}^{{\mu}_{n}}$, where $c\in R$ and ${\mu}_{1},\cdots ,{\mu}_{n}$ are integers $\ge 0$.

We consider polynomial mappings (f,g) of ℂ² with constant nontrivial jacobian. Using the Riemann-Hurwitz relation we prove among other things the following: If g - c (resp. f - c) has at most two branches at infinity for infinitely many numbers c or if f (resp. g) is proper on the level set ${g}^{-1}\left(0\right)$ (resp. ${f}^{-1}\left(0\right)$), then (f,g) is bijective.