Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polynomial Hessians with small rank   In this paper, the results in [Singular Hessians, J. Algebra 282 (2004), no.
1, 195--204], for polynomial Hessians with determinant zero in small dimensions
$r+1$, are generalized to similar results in arbitrary dimension, for
polynomial Hessians with rank $r$. All of this is over a field $K$ of
characteristic zero. The results in [Singular Hessians, J. Algebra 282 (2004),
no. 1, 195--204] are also reproved in a different perspective.
  One of these results is the classification by Gordan and Noether of
homogeneous polynomials in $5$ variables, for which the Hessians determinant is
zero. This result is generalized to homogeneous polynomials in general, for
which the Hessian rank is 4. Up to a linear transformation, such a polynomial
is either contained in $K[x_1,x_2,x_3,x_4]$, or contained in $$
K[x_1,x_2,p_3(x_1,x_2)x_3+p_4(x_1,x_2)x_4+\cdots+p_n(x_1,x_2)x_n] $$ for
certain $p_3,p_4,\ldots,p_n \in K[x_1,x_2]$ which are homogeneous of the same
degree.
  Furthermore, a new result which is similar to those in [Singular Hessians, J.
Algebra 282 (2004), no. 1, 195--204], is added, namely about polynomials $h \in
K[x_1,x_2,x_3,x_4,x_5]$, for which the last four rows of the Hessian matrix of
$t h$ are dependent. Here, $t$ is a variable, which is not one of those with
respect to which the Hessian is taken. This result is generalized to arbitrary
dimension as well: the Hessian rank of $t h$ is $4$ and the first row of the
Hessian matrix of $t h$ is independent of the other rows.

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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an