We study Newton polytopes for cluster variables in cluster algebras A(Σ) of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster variable can be written as a Laurent polynomial in the cluster variables of the initial seed Σ. The cluster variable Newton polytopes are the Newton polytopes of these Laurent polynomials. We show that if Σ has principal coefficients or boundary frozen variables, then all cluster variable Newton polytopes are saturated. We also characterize when these Newton polytopes are empty; that is, when they have no non-vertex lattice points.
Cluster algebras are rings with distinguished generators called cluster variables, grouped into clusters. Each cluster variable can be written as a Laurent polynomial in any cluster. In this paper, we study the Newton polytopes of these Laurent polynomials. We focus on cluster algebras of types A and D, with frozen variables corresponding to the boundary segments of a polygon and punctured polygon, respectively. For these cluster algebras, we show the cluster variable Newton polytopes are saturated. For type A, we additionally show that the cluster variable Newton polytopes have no non-vertex lattice points. Our main tool is the snake graph expansion formula of Musiker-Schiffler-Williams for cluster algebras from surfaces.
The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces P 1 × P 1 , we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have this property.
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