Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Candidates for non-zero Betti numbers of monomial ideals   Let $I$ be a monomial ideal in the polynomial ring $S$ generated by elements
of degree at most $d$. In this paper, it is shown that, if the $i$-th syzygy of
$I$ has no element of degrees $j, \ldots, j+(d-1)$ (where $j \geq i+d$), then
$(i+1)$-syzygy of $I$ does not have any element of degree $j+d$. Then we give
several applications of this result, including an alternative proof for
Green-Lazarsfeld index of the edge ideals of graphs as well as an alternative
proof for Fr\"oberg's theorem on classification of square-free monomial ideals
generated in degree two with linear resolution. Among all, we describe the
possible indices $i, j$ for which $I$ may have non-zero Betti numbers
$\beta_{i,j}$.

6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas