Related papers: Freiman ideals
An equigenerated monomial ideal $I$ is a Freiman ideal if $\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $\mu(I)$ is the least number of monomial generators of $I$. Freiman ideals are special…
An equigenerated monomial ideal $I$ in the polynomial ring $S= K[x_1,\ldots,x_n]$ is a Freiman ideal if $\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal…
A quasi-equigenerated monomial ideal $I$ in the polynomial ring $R= k[x_1, \ldots, x_n]$ is a Freiman ideal if $\mu(I^2) = l(I)\mu(I)- \binom{l(I)}{2}$ where $l(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal…
An equigenerated monomial ideal $I$ in the polynomial ring $R=k[z_1,\ldots, z_n]$ is a Freiman ideal if $\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal…
Freiman's theorem gives a lower bound for the cardinality of the doubling of a finite set in $\RR^n$. In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial…
Given any equigenerated monomial ideal $I$ with the property that the defining ideal $J$ of the fiber cone $ F(I)$ of $I$ is generated by quadratic binomials, we introduce a matrix such that the set of its binomial $2$-minors is a…
This paper was motivated by a problem left by Herzog and Hibi, namely to classify all unmixed polymatroidal ideals. In the particular case of polymatroidal ideals corresponding to discrete polymatroids of Veronese type, i.e ideals of…
Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…
For a monomial ideal $I$, let $G(I)$ be its minimal set of monomial generators. If there is a total order on $G(I)$ such that the corresponding Lyubeznik resolution of $I$ is a minimal free resolution of $I$, then $I$ is called a Lyubeznik…
In this paper it is shown that a sortable ideal $I$ is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese type ideals…
We study ideals which are generated by monomials of degree $d$ in the polynomial ring in $n$ variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese…
Let $k$ be a field of positive characteristic and $R = k[x_0,\dots, x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\geq \mathrm{height}(I)/d$; we…
We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a…
We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximal height of its minimal primes.
For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the…
We classify the minimal number of generators of artinian equigenerated monomial ideals $I$ such that $\Bbbk[x_1,\ldots,x_n]/I$ is forced to have the weak Lefschetz property.
We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for…
This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $\mu (I^2)\ge 9$ for the number of minimal…
We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also…
We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the…