Let F = (F₁,...,Fₙ): ℂⁿ → ℂⁿ be a polynomial mapping. By the multidegree of F we mean mdeg F = (deg F₁, ..., deg Fₙ) ∈ ℕ ⁿ. The aim of this paper is to study the following problem (especially for n = 3): for which sequence (d₁,...,dₙ) ∈ ℕ ⁿ is there a tame automorphism F of ℂⁿ such that mdeg F = (d₁,..., dₙ)? In other words we investigate the set mdeg(Tame(ℂⁿ)), where Tame(ℂⁿ) denotes the group of tame automorphisms of ℂⁿ.
Since mdeg(Tame(ℂⁿ)) is invariant under permutations of coordinates, we may...
Let V, W be algebraic subsets of , respectively, with n ≤ m. It is known that any finite polynomial mapping f: V → W can be extended to a finite polynomial mapping The main goal of this paper is to estimate from above the geometric degree of a finite extension of a dominating mapping f: V → W, where V and W are smooth algebraic sets.
Let d₃ ≥ p₂ > p₁ ≥ 3 be integers such that p₁,p₂ are prime numbers. We show that the sequence (p₁,p₂,d₃) is the multidegree of some tame automorphism of ℂ³ if and only if d₃ ∈ p₁ℕ + p₂ℕ, i.e. if and only if d₃ is a linear combination of p₁ and p₂ with coefficients in ℕ.
We prove that every locally nilpotent monomial k-derivation of k[X₁,...,Xₙ] is triangular, whenever k is a ring of characteristic zero. A method of testing monomial k-derivations for local nilpotency is also presented.
We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, , and such that and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.
Assume that X,Y are integral noetherian affine schemes. Let f:X → Y be a dominant, generically finite morphism of finite type. We show that the set of points at which the morphism f is not finite is either empty or a hypersurface. An example is given to show that this is no longer true in the non-noetherian case.
Let d be any integer greater than or equal to 3. We show that the intersection of the set mdeg(Aut(ℂ³))∖ mdeg(Tame(ℂ³)) with {(d₁,d₂,d₃) ∈ (ℕ ₊)³: d = d₁ ≤ d₂≤ d₃} has infinitely many elements, where mdeg h = (deg h₁,...,deg hₙ) denotes the multidegree of a polynomial mapping h = (h₁,...,hₙ): ℂⁿ → ℂⁿ. In other words, we show that there are infinitely many wild multidegrees of the form (d,d₂,d₃), with fixed d ≥ 3 and d ≤ d₂ ≤ d₃, where a sequence (d₁,...,dₙ)∈ ℕ ⁿ is a wild multidegree if there is...
In this paper, we discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research was based on, essentially, one of the most important results in mathematical finance, called the Fundamental Theorem of Asset Pricing. For the standard approach a risk-free bank account process is used as numeraire. In those models it is assumed that the interest rates for borrowing and saving money are the same. In our paper we consider the...
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