Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSymmetric Polynomials in Tropical Algebra Semirings
67
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max, +), Izhakian-Rowens extended and supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. This allows us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.
rate research
Read More
Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets n powersums forming a regular sequence. A necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a<b<c with GCD(a,b,c)=1, we conjecture that p_a, p_b, p_c is a regular sequence for n=3 if and only if 6 divides abc. We provide evidence for the conjecture by proving it in several special instances.
We study the symmetric subquotient decomposition of the associated graded algebras $A^*$ of a non-homogeneous commutative Artinian Gorenstein (AG) algebra $A$. This decomposition arises from the stratification of $A^*$ by a sequence of ideals $A^*=C_A(0)supset C_A(1)supsetcdots$ whose successive quotients $Q(a)=C(a)/C(a+1)$ are reflexive $A^*$ modules. These were introduced by the first author, and have been used more recently by several groups, especially those interested in short Gorenstein algebras, and in the scheme length (cactus rank) of forms. For us a Gorenstein sequence is an integer sequence $H$ occurring as the Hilbert function for an AG algebra $A$, that is not necessarily homogeneous. Such a Hilbert function $H(A)$ is the sum of symmetric non-negative sequences $H_A(a)=H(Q_A(a))$, each having center of symmetry $(j-a)/2$ where $j$ is the socle degree of $A$: we call these the symmetry conditions, and the decomposition $mathcal{D}(A)=(H_A(0),H_A(1),ldots)$ the symmetric decomposition of $H(A)$. We here study which sequences may occur as the summands $H_A(a)$: in particular we construct in a systematic way examples of AG algebras $A$ for which $H_A(a)$ can have interior zeroes, as $H_A(a)=(0,s,0,ldots,0,s,0)$. We also study the symmetric decomposition sets $mathcal{D}(A)$, and in particular determine which sequences $H_A(a)$ can be non-zero when the dual generator is linear in a subset of the variables. Several groups have studied exotic summands of the Macaulay dual generator $F$. Studying these, we recall a normal form for the Macaulay dual generator of an AG algebra that has no exotic summands. We apply this to Gorenstein algebras that are connected sums. We give throughout many examples and counterexamples, and conclude with some open questions about symmetric decomposition.
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.
Differential graded (DG) algebras are powerful tools from rational homotopy theory. We survey some recent applications of these in the realm of homological commutative algebra.
We investigate geometric embeddings among several classes of stacky fans and algorithms, e.g., to compute their homology. Interesting cases arise from moduli spaces of tropical curves. Specifically, we study the embedding of the moduli of tropical honeycomb curves into the moduli of all tropical $K_4$-curves.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
