The algebra of the box spline

In this paper we want to revisit results of Dahmen and Micchelli on box-splines which we reinterpret and make more precise. We compare these ideas with the work of Brion, Szenes, Vergne and others on polytopes and partition functions.Comment: 69 page

Pure Differential Modules and a Result of Macaulay on Unmixed Polynomial Ideals

The first purpose of this paper is to point out a curious result announced by Macaulay on the Hilbert function of a differential module in his famous book The Algebraic Theory of Modular Systems published in 1916. Indeed, on page 78/79 of this book, Macaulay is saying the following: " A polynomial ideal $\mathfrak{a} \subset k[{\chi}\_1$,..., ${\chi}\_n]=k[\chi]$ is of the {\it principal class} and thus {\it unmixed} if it has rank $r$ and is generated by $r$ polynomials. Having in mind this definition, a primary ideal $\mathfrak{q}$ with associated prime ideal $\mathfrak{p} = rad(\mathfrak{q})$ is such that any ideal $\mathfrak{a}$ of the principal class with $\mathfrak{a} \subset \mathfrak{q}$ determines a primary ideal of greater {\it multiplicity} over $k$. In particular, we have $dim\_k(k[\chi]/({\chi}\_1$,...,${\chi}\_n)^2)=n+1$ because, passing to a system of PD equations for one unknown $y$, the parametric jets are \{${y,y\_1, ...,y\_n}$\} but any ideal $\mathfrak{a}$ of the principal class with $\mathfrak{a}\subset ({\chi}\_1,{\^a},{\chi}\_n)^2$ is contained into a {\it simple} ideal, that is a primary ideal $\mathfrak{q}$ such that $rad(\mathfrak{q})=\mathfrak{m}\in max(k[\chi])$ is a maximal and thus prime ideal with $dim\_k(M)=dim\_k(k[\chi]/\mathfrak{q})=2^n$ at least. Accordingly, any primary ideal $\mathfrak{q}$ may not be a member of the primary decomposition of an unmixed ideal $\mathfrak{a} \subseteq \mathfrak{q}$ of the principal class. Otherwise, $\mathfrak{q}$ is said to be of the {\it principal noetherian class} ". Our aim is to explain this result in a modern language and to illustrate it by providing a similar example for $n=4$. The importance of such an example is that it allows for the first time to exhibit symbols which are $2,3,4$-acyclic without being involutive. Another interest of this example is that it has properties quite similar to the ones held by the system of conformal Killing equations which are still not known. For this reason, we have put all the examples at the end of the paper and each one is presented in a rather independent way though a few among them are quite tricky. Meanwhile, the second purpose is to prove that the methods developped by Macaulay in order to study {\it unmixed polynomial ideals} are only particular examples of new formal differential geometric techniques that have been introduced recently in order to study {\it pure differential modules}. However these procedures are based on the formal theory of systems of ordinary differential (OD) or partial differential (PD) equations, in particular on a systematic use of the Spencer operator, and are still not acknowledged by the algebraic community

Macaulay inverse systems revisited

Since its original publication in 1916 under the title "The Algebraic Theory of Modular Systems", the book by F. S. Macaulay has attracted a lot of scientists with a view towards pure mathematics (D. Eisenbud,...) or applications to control theory (U. Oberst,...).However, a carefull examination of the quotations clearly shows that people have hardly been looking at the last chapter dealing with the so-called "inverse systems", unless in very particular situations. The purpose of this paper is to provide for the first time the full explanation of this chapter within the framework of the formal theory of systems of partial differential equations (Spencer operator on sections, involution,...) and its algebraic counterpart now called "algebraic analysis" (commutative and homological algebra, differential modules,...). Many explicit examples are fully treated and hints are given towards the way to work out computer algebra packages.Comment: From a lecture at the International Conference : Application of Computer Algebra (ACA 2008) july 2008, RISC, LINZ, AUSTRI

Feynman integral relations from parametric annihilators

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space IBP relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional remark