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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 mdegh=(degh1,…,deghn) 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≤d...
Let d3≥ p2> p1≥ 3 be integers such that p1, p2are prime numbers. We show that the sequence (p1, p2, d3) is the multidegree of some tame automorphism of C3if and only if d3∈ p1N p2N, i.e. if and only if d3is a linear combination of p1and p2with coefficients in N.
We present an example of finite mappings of algebraic varieties ƒ : V → W, where V ⊂ kn, W ⊂ kn+1, and F : kn → kn+1 such that F¦v = ƒ and gdeg F = 1 < gdeg/ (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 ⊂ km with...
Let ƒ : V → W be a finite polynomial mapping of algebraic subsets V, W of k[sup]n and k[sup]m, respectively, with n ≤ m. Kwieciński [J. Pure Appl. Algebra 76 (1991)] proved that there exists a finite polynomial mapping F : k[sup]n → k[sup]m such that F| v = ƒ. In this note we prove that, if V, W ⊂ k[sup] are smooth of dimension k with 3k + 2 ≤ n, and ƒ : V → W is finite, dominated and dominated on...
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.
In this note we prove that for every finite sets V, W [is a subset of] [C^k] with k, #V, #W > 1 and for every surjective mapping f : V --> W there exists a finite mapping F : [C^k] --> [C^k] such that F\v = f, gdegF = gdegf and degF [is less than or equal to (#V - 1)^2].
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