Weak Bourbaki unmixed rings are defined in this paper. The definition of a weak Bourbaki unmixed ring is a candidate for an “appropriate” definition of CohenMacaulayness. We will see that this definition satisfies many of the conditions we want an “appropriate” definition to satisfy. It is not yet known whether this definition (or any other) satisfies all of the conditions. However, no example ...

A ΣΠΣΠ(k) circuit C = ∑k i=1 Fi = ∑k i=1 ∏di j=1 fij is unmixed if for each i ∈ [k], Fi = fi1(x1) · · · fin(xn), where each fij is a univariate polynomial given in the sparse representation. In this paper, we give a polynomial time black-box algorithm of identity testing for the low degree unmixed ΣΠΣΠ(k) circuits. In order to obtain the black-box algorithm, we first show that a special class o...

Symmetric edge polytopes, a.k.a. PV-type adjacency associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, dynamical systems. In particular, the authors are motivated by study of algebraic Kuramoto equations unmixed form whose Newton polytopes symmetric polytopes. The interplay between geometric structure...

A ΣΠΣΠ(k) circuit C = ∑k i=1 Fi = ∑k i=1 ∏di j=1 fij is unmixed if for each i ∈ [k], Fi = fi1(x1) · · · fin(xn), where each fij is a univariate polynomial given in the sparse representation. In this paper, we give a polynomial time black-box algorithm of identity testing for the low degree unmixed ΣΠΣΠ(k) circuits. In order to obtain the black-box algorithm, we first show that a special class o...