We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.
We compute the resultants for quadratic binomial complete intersections. As an application we show that any quadratic binomial complete intersection can have the set of square-free monomials as a vector space basis if the generators are put in a normal form.
The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(n\times n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omitted. If $I_n$ is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights into binomial ideals and multigraded algebra. For example, if $n$ is even, $I_n$ is a Gorenstein ideal and the intersection of the remaining primary components of $J_n$ equals $J_n+\lan…
We classify all binomial edge ideals that are complete intersection and Cohen-Macaulay almost complete intersection. We also describe an algorithm and provide an implementation to compute primary decomposition of binomial edge ideals.
Using aritmethic conditions on affine semigroups we prove that for a simplicial toric variety of codimension 2 the property of being a set-theoretic complete intersection on binomials in characteristic $p$ holds either for all primes $p$, or for no prime $p$, or for exactly one prime $p$.
The binomial ideal associated with the intersection axiom of conditional probability is shown to be radical and is expressed as intersection of toric prime ideals. This resolves a conjecture in algebraic statistics due to Cartwright and Engström.
Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the second graded Betti numbers of $J_G$, and we obtain a minimal presentation of it when $G$ is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a $d$-sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees …
In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be suff…
In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an almost complete intersection ideal $I$ which is generated by a homogeneous $d$-sequence. As applications, when $I$ is an almost complete intersection, taking the form of the (parity) binomial edge ideal of a connected graph, we can describe explicitly formulas for $\reg(I^t)$ for $t\ge 2$. The only exception is when $I$ is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.
In this paper, we study a family of binomial ideals defining monomial curves in the $n-$dimensional affine space determined by $n$ hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}} \in k[x_1, \ldots, x_n]$ with $u_{ii} = 0$, $i\in \{ 1, \ldots, n\}$. We prove that, the monomial curves in that family are set-theoretic complete intersection. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Fröbenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semig…
We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming.
We present a class of toric varieties $V$ which, over any algebraically closed field of characteristic zero, are defined by codim $V$+1 binomial equations.
In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of $\bz^n\oplus T$ with no invertible elements, where $T$ is a finite abelian group. We also characterize the lattice ideals that are set-theoretic complete intersections on binomials.
In this paper, we study the strong Lefschetz property of artinian complete intersection ideals generated by products of linear forms. We prove the strong Lefschetz property for a class of such ideals with binomial generators.
We describe a class of affine toric varieties $V$ that are set-theoretically minimally defined by codim $V+1$ binomial equations over fields of any characteristic.
Let $X$ be the Hankel matrix of size $2\times n$ and let $G$ be a closed graph on the vertex set $[n].$ We study the binomial ideal $I_G\subset K[x_1,\ldots,x_{n+1}]$ which is generated by all the $2$-minors of $X$ which correspond to the edges of $G.$ We show that $I_G$ is Cohen-Macaulay. We find the minimal primes of $I_G$ and show that $I_G$ is a set theoretical complete intersection. Moreover, a sharp upper bound for the regularity of $I_G$ is given.
We present a class of homogeneous ideals which are generated by monomials and binomials of degree two and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the presentation ideals of the fiber cone algebras of monomial varieties of codimension two.
We present examples which show that in dimension higher than one or codimension higher than two, there exist toric ideals I_A such that no binomial ideal contained in I_A and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations.
Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.
Let $R = S/I$ be a quotient of a standard graded polynomial ring $S$ by an ideal $I$ generated by quadrics. If $R$ is Koszul, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R$ over $S$ can be bounded above by binomial coefficients on the minimal number of generators of $I$. Motivated by previous results for Koszul algebras defined by three quadrics, we give a complete classification of the structure of Koszul almost complete intersections and, in the process, give an affirmative answer to the above question for all such rings.
For a discrete valuation domain $V$ with maximal ideal $\mathfrak{m}$ such that the residue field $V/\mathfrak{m}$ is finite, there exists a sequence of polynomials $(F_n(x))_{n \ge 0}$ defined over the quotient field $K$ of $V$ that forms a basis of the $V$-module $\text{Int}(V) = \{f \in K[x] | f(V)\subseteq V\}$. This sequence of polynomials bears many resemblances to the classical binomial polynomials $(\binom{x}{n})_{n \ge 0}$. In this paper, we introduce a generating polynomial to account for the distribution of the $V$-values of the polynomials $F_n(x)$ modulo the maximal ideal $\mathfr…
Waring rank of certain complete intersection forms
Nanyang Technological University（南洋理工大学）；Politecnico Di Torino（都灵理工大学）；Indian Institute of Technology Dharwad（达尔瓦德印度理工学院）；Indian Institute of Technology Bombay（孟买印度理工学院）
Let $G=(V,E)$ be a finite, simple graph. We consider for each oriented graph $G_{\cal O}$ associated to an orientation ${\cal O}$ of the edges of $G$, the toric ideal $P_{G_{\cal O}}$. In this paper we study those graphs with the property that $P_{G_{\cal O}}$ is a binomial complete intersection, for all ${\cal O}$. These graphs are called $\text{CI}{\cal O}$ graphs. We prove that these graphs can be constructed recursively as clique-sums of cycles and/or complete graphs. We introduce the chorded-theta subgraphs and their transversal triangles. Also we establish that the $\text{CI}{\cal O}$ gr…
We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In particular, we provide explicit counting formulas that have also applications to some Hermitian codes.
