Volume 30, Issue 4
The Dependence Problem for a Class of Polynomial Maps in Dimension Four

Commun. Math. Res., 30 (2014), pp. 289-294.

Published online: 2021-05

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• Abstract

Let $h$ be a polynomial in four variables with the singular Hessian $\mathcal{H}h$ and the gradient $∇h$ and $R$ be a nonzero relation of $∇h$. Set $H = ∇R(∇h)$. We prove that the components of $H$ are linearly dependent when $rk\mathcal{H}h ≤ 2$ and give a necessary and sufficient condition for the components of $H$ to be linearly dependent when $rk\mathcal{H}h = 3$.

• Keywords

dependence problem, linear dependence, quasi-translation.

• AMS Subject Headings

14R99

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@Article{CMR-30-289, author = {Yong and Jin and and 18862 and and Yong Jin and Hongbo and Guo and and 18863 and and Hongbo Guo}, title = {The Dependence Problem for a Class of Polynomial Maps in Dimension Four}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {30}, number = {4}, pages = {289--294}, abstract = {

Let $h$ be a polynomial in four variables with the singular Hessian $\mathcal{H}h$ and the gradient $∇h$ and $R$ be a nonzero relation of $∇h$. Set $H = ∇R(∇h)$. We prove that the components of $H$ are linearly dependent when $rk\mathcal{H}h ≤ 2$ and give a necessary and sufficient condition for the components of $H$ to be linearly dependent when $rk\mathcal{H}h = 3$.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2014.04.01}, url = {http://global-sci.org/intro/article_detail/cmr/18953.html} }
TY - JOUR T1 - The Dependence Problem for a Class of Polynomial Maps in Dimension Four AU - Jin , Yong AU - Guo , Hongbo JO - Communications in Mathematical Research VL - 4 SP - 289 EP - 294 PY - 2021 DA - 2021/05 SN - 30 DO - http://doi.org/10.13447/j.1674-5647.2014.04.01 UR - https://global-sci.org/intro/article_detail/cmr/18953.html KW - dependence problem, linear dependence, quasi-translation. AB -

Let $h$ be a polynomial in four variables with the singular Hessian $\mathcal{H}h$ and the gradient $∇h$ and $R$ be a nonzero relation of $∇h$. Set $H = ∇R(∇h)$. We prove that the components of $H$ are linearly dependent when $rk\mathcal{H}h ≤ 2$ and give a necessary and sufficient condition for the components of $H$ to be linearly dependent when $rk\mathcal{H}h = 3$.

Yong Jin & Hongbo Guo. (2021). The Dependence Problem for a Class of Polynomial Maps in Dimension Four. Communications in Mathematical Research . 30 (4). 289-294. doi:10.13447/j.1674-5647.2014.04.01
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