Computing Algebraic Matroids

An affine variety induces the structure of an algebraic matroid on the set of coordinates of the ambient space. The matroid has two natural decorations: a circuit polynomial attached to each circuit, and the degree of the projection map to each base, called the base degree. Decorated algebraic matroids can be computed via symbolic computation using Groebner bases, or through linear algebra in the space of differentials (with decorations calculated using numerical algebraic geometry). Both algorithms are developed here. Failure of the second algorithm occurs on a subvariety called the non-matroidal or NM- locus. Decorated algebraic matroids have widespread relevance anywhere that coordinates have combinatorial significance. Examples are computed from applied algebra, in algebraic statistics and chemical reaction network theory, as well as more theoretical examples from algebraic geometry and matroid theory.Comment: 15 pages; added link to references, note on page 1, and small formatting fixe

Algebraic matroids with graph symmetry

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely

The Twilight of the Opera Pirates: A Prehistory of the Exclusive Right of Public Performance for Musical Compositions

The exclusive right of public performance of a musical composition now brings to composers and songwriters revenue of approximately one billion dollars a year in the US alone. However, this right was not firmly established until a century after America’s first copyright statute, relying until then on the common-law principles that protected unpublished works. The first effort to create this right by statute was the Ingersoll Copyright Bill, an omnibus revision in 1844 which died quickly in committee. After that 50 years passed, and in the final quarter of the nineteenth century the need for statutory protection for public performance became more and more obvious as a result of litigation, especially that surrounding the Gilbert and Sullivan operetta The Mikado. In the mid-1890s the right was once again proposed in an omnibus revision that died in committee, the Treloar Copyright Bill. Simultaneously though, this right went through Congress and was passed as part of an amendatory act which also increased penalties for all unlawful public performances (including drama). This article traces the history of these acts and the litigation in the later nineteenth century, telling a story that has heretofore not been told – the prehistory of the right of public of public performance for musical compositions

How Perris v. Hexamer Was Lost in the Shadow of Baker v. Selden

Perris v. Hexamer stands out as case that is equal parts important and forgotten. It is obviously important – it is one of a preciously small number of Supreme Court decisions on the idea/expression dichotomy, but it is mostly forgotten in favor of the Court’s decision the following year in Baker v. Selden. It is equally obscure – Westlaw counts 2,703 citations of Baker v. Selden, and 81 of Perris v. Hexamer. Yet the subject matter of both decisions is surprisingly similar, and these cases tell us far more when considered in tandem than when either one is considered on its own. This piece will seek to tell the story of Perris v. Hexamer – in terms of both the background of the controversy and the procedural background that led to the lawsuit, as well as discussing the decision itself. Following this, two questions will be addressed – firstly why Perris was largely forgotten as a decision about the idea/expression dichotomy, and secondly why the vote among the Justices was different in Perris than in Baker v. Selden