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Multidegrees of tame automorphisms of ℂⁿ

100%
Karaś M.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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 focus on the set {(d₁,...,dₙ): d₁ ≤ ⋯ ≤ dₙ} ∩ mdeg (Tame(ℂⁿ)). Obviously, we have {(1,d₂,d₃): 1 ≤ d₂ ≤ d₃} ∩ mdeg(Tame(ℂ³)) = {(1,d₂,d₃): 1 ≤ d₂ ≤ d₃}. Not obvious, but still easy to prove is the equality mdeg(Tame(ℂ³)) ∩ {(2,d₂,d₃): 2 ≤ d₂ ≤ d₃} = {(2,d₂,d₃): 2 ≤ d₂ ≤ d₃}. We give a complete description of the sets {(3,d₂,d₃): 3 ≤ d₂ ≤ d₃} ∩ mdeg(Tame(ℂ³)) and {(5,d₂,d₃): 5 ≤ d₂ ≤ d₃} ∩ mdeg(Tame(ℂ³)). In the examination of the last set the most difficult part is to prove that (5,6,9) ∉ mdeg(Tame(ℂ³)). To do this, we use the two-dimensional Jacobian Conjecture (which is true for low degrees) and the Jung-van der Kulk Theorem. As a surprising consequence of the method used in proving that (5,6,9) ∉ mdeg(Tame(ℂ³)), we show that the existence of a tame automorphism F of ℂ³ with mdeg F = (37,70,105) implies that the two-dimensional Jacobian Conjecture is not true. Also, we give a complete description of the following sets: {(p₁,p₂,d₃): 2 < p₁ < p₂ ≤ d₃, p₁,p₂ prime numbers} ∩ mdeg(Tame(ℂ³)), {(d₁,d₂,d₃): d₁ ≤ d₂ ≤ d₃, d₁,d₂ ∈ 2ℕ +1, gcd(d₁,d₂) = 1} ∩ mdeg(Tame(ℂ³)). Using the description of the last set we show that mdeg(Aut(ℂ³))∖ mdeg(Tame(ℂ³)) is infinite. We also obtain a (still incomplete) description of the set mdeg(Tame(ℂ³)) ∩ {(4,d₂,d₃): 4 ≤ d₂ ≤ d₃} and we give complete information about $mdeg F^{-1}$ for F ∈ Aut(ℂ²).
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Tame Automorphisms of ℂ³ with Multidegree of the Form (p₁,p₂,d₃)

95%
Karaś M.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
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tom 59
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nr 1
27-32
EN
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.
3
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Birational Finite Extensions of Mappings from a Smooth Variety

95%
Karaś M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2009
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tom 57
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nr 2
117-120
EN
We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, $W ⊂ k^{n+1}$, and $F:kⁿ → k^{n+1}$ such that $F|_{V} = f$ 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, $W ⊂ k^{m}$ with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.
4
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A Note on Geometric Degree of Finite Extensions of Mappings from a Smooth Variety

95%
Karaś M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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tom 56
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nr 2
105-108
EN
Let f:V → W be a finite polynomial mapping of algebraic subsets V,W of kⁿ and $k^{m}$, respectively, with n ≤ m. Kwieciński [J. Pure Appl. Algebra 76 (1991)] proved that there exists a finite polynomial mapping $F:kⁿ → k^{m}$ such that $F|_{V} = f$. In this note we prove that, if V,W ⊂ kⁿ are smooth of dimension k with 3k+2 ≤ n, and f:V → W is finite, dominated and dominated on every component, then there exists a finite polynomial mapping F: kⁿ → kⁿ$ such that $F|_{V} = f$ and $gdeg F ≤ (gdeg f)^{k+1}$. This improves earlier results of the author.
5
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Locally Nilpotent Monomial Derivations

95%
Karaś M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2004
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tom 52
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nr 2
119-121
EN
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.
6
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Finite extensions of mappings from a smooth variety

95%
Karaś M.
Annales Polonici Mathematici
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2000
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tom 75
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nr 1
79-86
EN
Let V, W be algebraic subsets of $k^n$, $k^m$ respectively, with n ≤ m. It is known that any finite polynomial mapping f: V → W can be extended to a finite polynomial mapping $F: k^{n} → k^{m}.$ The main goal of this paper is to estimate from above the geometric degree of a finite extension $F: k^n → k^n$ of a dominating mapping f: V → W, where V and W are smooth algebraic sets.
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Discrete-time market models from the small investor point of view and the first fundamental-type theorem

61%
Karaś M., Serwatka A.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2017
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tom 16
EN
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 model of a market (with d risky assets), which does not hold the same interest rate assumptions. We introduce two predictable processes for modelling deposits and loans. We propose a new concept of a martingale pair for the market and prove that if there exists a martingale pair for the considered market, then there is no arbitrage opportunity. We also consider special cases in which the existence of a martingale pair is necessary and the sufficient conditions for these markets to be arbitrage free.
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Wild Multidegrees of the Form (d,d₂,d₃) for Fixed d ≥ 3

61%
Karaś M., Zygadło J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2012
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tom 60
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nr 3
211-218
EN
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 a polynomial automorphism F of ℂⁿ with mdeg F = (d₁,...,dₙ), and there is no tame automorphism of ℂⁿ with the same multidegree.
9
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The set of points at which a morphism of affine schemes is not finite

61%
Jelonek Z., Karaś M.
Colloquium Mathematicum
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2002
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tom 92
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nr 1
59-66
EN
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.
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