142 research outputs found
Computing in algebraic geometry and commutative algebra using Macaulay 2
AbstractWe present recent research of Eisenbud, Fløystad, Schreyer, and others, which was discovered with the help of experimentation with Macaulay 2. In this invited, expository paper, we start by considering the exterior algebra, and the computation of Gröbner bases. We then present, in an elementary manner, the explicit form of the Bernstein–Gelfand–Gelfand relationship between graded modules over the polynomial ring and complexes over the exterior algebra, that Eisenbud, Fløystad and Schreyer found. We present two applications of these techniques: cohomology of sheaves, and the construction of determinantal formulae for (powers of) Macaulay resultants. We show how to use Macaulay 2 to perform these computations
Computing periods of rational integrals
A period of a rational integral is the result of integrating, with respect to
one or several variables, a rational function over a closed path. This work
focuses particularly on periods depending on a parameter: in this case the
period under consideration satisfies a linear differential equation, the
Picard-Fuchs equation. I give a reduction algorithm that extends the
Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs
equations. The resulting algorithm is elementary and has been successfully
applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at
http://pierre.lairez.fr/supp/periods
Polynomial Invariants for Affine Programs
We exhibit an algorithm to compute the strongest polynomial (or algebraic)
invariants that hold at each location of a given affine program (i.e., a
program having only non-deterministic (as opposed to conditional) branching and
all of whose assignments are given by affine expressions). Our main tool is an
algebraic result of independent interest: given a finite set of rational square
matrices of the same dimension, we show how to compute the Zariski closure of
the semigroup that they generate
Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves
We show that for a class of two-loop diagrams, the on-shell part of the
integration-by-parts (IBP) relations correspond to exact meromorphic one-forms
on algebraic curves. Since it is easy to find such exact meromorphic one-forms
from algebraic geometry, this idea provides a new highly efficient algorithm
for integral reduction. We demonstrate the power of this method via several
complicated two-loop diagrams with internal massive legs. No explicit elliptic
or hyperelliptic integral computation is needed for our method.Comment: minor changes: more references adde
Solving polynomial systems via symbolic-numeric reduction to geometric involutive form
AbstractWe briefly survey several existing methods for solving polynomial systems with inexact coefficients, then introduce our new symbolic-numeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method
Maurice Janet's algorithms on systems of linear partial differential equations
This article presents the emergence of formal methods in theory of partial differential equations (PDE) in the french school of mathematics through Janet's work in the period 1913-1930. In his thesis and in a series of articles published during this period, M. Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the unicity of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. M. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the 20th century in various algebraic contexts
Hochschild Cohomology Of Short Gorenstein Rings
Let k be a field of characteristic 0. In this thesis, we show that the Hochschild cohomology of the family of short Gorenstein k-algebras
sGor(N) =k[X_0,...,X_N](X_iX_j, X_i^2−X_j^2 | i,j= 0,...,N, i different from j), N≥2,
exhibits exponential growth. The proof uses Gröbner-Shirshov basis theory and along the way we describe an explicit monomial basis for the Koszul dual of sGor(N) for N≥2
Ideals generated by the inner 2-minors of collections of cells
In 2012 Ayesha Asloob Qureshi connected collections of cells to Commutative Algebra assigning to every collection of cells the ideal of inner 2-minors, denoted by , in the polynomial ring . Investigating the main algebraic properties of depending on the shape of is the purpose of this research. Many problems are still open and they seem to be fascinating and exciting challenges.\\
In this thesis we prove several results about the primality of and the algebraic properties of like Cohen-Macaulyness, normality and Gorensteiness, for some classes of non-simple polyominoes. The study of the Hilbert-Poincar\'e series and the related invariants as Krull dimension and Castelnuovo-Mumford regularity are given. Finally we provide the code of the package \texttt{PolyominoIdeals} developed for \texttt{Macaulay2}
- …