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Register a new userExponential ideals and a Nullstellensatz
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We prove a version of a Nullstellensatz for partial exponential fields $(K,E)$, even though the ring of exponential polynomials $K[X_1,ldots,X_n]^E$ is not a Hilbert ring. We show that under certain natural conditions one can embed an ideal of $K[X_1,ldots,X_n]^E$ into an exponential ideal. In case the ideal consists of exponential polynomials with one iteration of the exponential function, we show that these conditions can be met. We apply our results to the case of ordered exponential fields.
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Since a tropical Nullstellensatz fails even for tropical univariate polynomials we study a conjecture on a tropical {it dual} Nullstellensatz for tropical polynomial systems in terms of solvability of a tropical linear system with the Cayley matrix associated to the tropical polynomial system. The conjecture on a tropical effective dual Nullstellensatz is proved for tropical univariate polynomials.
The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $mathbb Q[x_1, dots, x_n]$ to a corresponding ideal in $mathbb F_p[x_1,dots, x_n]$ where $p$ is a prime number; in other words, the textit{reduction modulo $p$} of $I$. We first define a new notion of $sigma$-good prime for $I$ which does depends on the term ordering $sigma$, but not on the given generators of $I$. We relate our notion of $sigma$-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo~$p$ from the term ordering, thus letting us show that all but finitely many primes are good for $I$. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.
Let $G$ be a simple graph on $n$ vertices. Let $L_G text{ and } mathcal{I}_G : $ denote the Lovasz-Saks-Schrijver(LSS) ideal and parity binomial edge ideal of $G$ in the polynomial ring $S = mathbb{K}[x_1,ldots, x_n, y_1, ldots, y_n] $ respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we prove that their Rees algebra is Cohen-Macaulay. We compute the second graded Betti number and obtain a minimal presentation of LSS ideals of trees and odd unicyclic graphs. We also obtain an explicit description of the defining ideal of the symmetric algebra of LSS ideals of trees and odd unicyclic graphs.
Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In two previous papers we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.
What kind of reduced monomial schemes can be obtained as a Grobner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. We discuss consequences for rational and F-rational singularities of algebras with straightening laws.
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