arXiv subjects

Phan Thanh Toan

Publications and source records attributed to Phan Thanh Toan.

6 recordsLinked to original sources

arXiv2026

Automorphic Transcendence Degree, Growth, and Skew Normalization

We study automorphic transcendence degree for quotients of multivariate skew polynomial rings over a division ring D. We identify this invariant with relative growth dimension, the degree of the relative Hilbert polynomial, and a coordinate dimension determined by leading monomials. We establish invariance under finite module extensions, integral extensions, and investigate the Gelfand-Kirillov dimension under a local finiteness condition. When the coefficient automorphisms are independent modulo inner automorphisms, we prove that a quotient is automorphically normalizable precisely when its automorphic transcendence degree equals the number of nonnilpotent coordinate variables, and classify all such normalization subrings.

math.RA↗

arXiv2026

Polynomial extensions do not preserve the strong finite type property

Arnold introduced the strong finite type (SFT) property in 1973 while studying the dimension of power series rings. For several classes of rings, polynomial extension is known to preserve the SFT property, but the general question remained open. We answer it negatively by constructing an SFT ring $R$ such that $R[X]$ is not SFT.

math.AC↗

arXiv2026

Noether's normalization in iterated skew polynomial rings

The classical Noether Normalization Lemma states that if $S$ is a finitely generated algebra over a field $k$, then there exist elements $x_1,\dots,x_n$ which are algebraically independent over $k$ such that $S$ is a finite module over $k[x_1,\dots,x_n]$. This lemma has been studied intensively in different flavors. In 2024, Elad Paran and Thieu N. Vo successfully generalized this lemma for the case when $S$ is a quotient ring of the skew polynomial ring $D[x_1,\dots,x_n;σ_1,\dots,σ_n]$. In this paper, we investigate this lemma in a more general setting when $S$ is a quotient ring of an iterated skew polynomial ring $D[x_1;σ_1,δ_1]\dots[x_n;σ_n,δ_n]$. We extend several key results of Elad Paran and Thieu N. Vo to this broader context and introduce a new version of Combinatorial Nullstellensatz over division rings.

math.RA↗

arXiv2012

New Inequalities for q-ary Constant-Weight Codes

Using double counting, we prove Delsarte inequalities for $q$-ary codes and their improvements. Applying the same technique to $q$-ary constant-weight codes, we obtain new inequalities for $q$-ary constant-weight codes.

math.CO↗

arXiv2012

Improved Semidefinite Programming Bound on Sizes of Codes

Let $A(n,d)$ (respectively $A(n,d,w)$) be the maximum possible number of codewords in a binary code (respectively binary constant-weight $w$ code) of length $n$ and minimum Hamming distance at least $d$. By adding new linear constraints to Schrijver's semidefinite programming bound, which is obtained from block-diagonalising the Terwilliger algebra of the Hamming cube, we obtain two new upper bounds on $A(n,d)$, namely $A(18,8) \leq 71$ and $A(19,8) \leq 131$. Twenty three new upper bounds on $A(n,d,w)$ for $n \leq 28$ are also obtained by a similar way.

cs.IT↗

arXiv2011

Improved Linear Programming Bounds on Sizes of Constant-Weight Codes

Let $A(n,d,w)$ be the largest possible size of an $(n,d,w)$ constant-weight binary code. By adding new constraints to Delsarte linear programming, we obtain twenty three new upper bounds on $A(n,d,w)$ for $n \leq 28$. The used techniques allow us to give a simple proof of an important theorem of Delsarte which makes linear programming possible for binary codes.

cs.IT↗