Xiaosong Sun

Multidegrees of Tame Automorphisms in Dimension Three

We discuss when a sequence of positive integers can be the multidegree of some tame automorphism in dimension three, and we also relate these investigations to the problem of whether there exists a tame automorphism admitting a reduction of type II or type III. 2010 Mathematics Subject Classification: Primary 14R10.

Polynomial maps with invertible sums of Jacobian matrices and directional derivatives

Abstract Let F : C n → C m be a polynomial map with deg F = d ≥ 2 . We prove that F is invertible if m = n and ∑ i = 1 d − 1 ( J F ) | α i is invertible for all α i ∈ C n , which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines L = { β + μ γ ∣ μ ∈ C } ⊆ C n ( γ ≠ 0 ), F ∣ L is linearly rectifiable, if and only if ∑ i = 1 d − 1 ( J F ) | α i ⋅ γ ≠ 0 for all α i ∈ L . This appears to be the case for all affine lines L when F is injective and d ≤ 3 . We also prove that if m = n and ∑ i = 1 n ( J F ) | α i is invertible for all α i ∈ C n , then F is a composition of an invertible linear map and an invertible polynomial map X + H with linear part X , such that the subspace generated by { ( J H ) | α ∣ α ∈ C n } consists of nilpotent matrices.

Images of higher-order differential operators of polynomial algebras

We investigate images of higher-order differential operators of polynomial algebras over a field

On non-singular multilinear maps

k

On the range of a Hadamard power of a positive semidefinite matrix

. We show that, when

On quadratic homogeneous quasi-translations

\operatorname{char}k>0

Quadratic linear Keller maps of nilpotency index three

, the image of the set of differential operators

The linear dependence problem for power linear maps

\{\unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D70F}_{i}\mid i=1,2,\ldots ,n\}

Quadratic homogeneous polynomial maps

of the polynomial algebra

H

k[\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{n},z_{1},\ldots ,z_{n}]