The Construction of Noetherian Operators

Oberst, Ulrich

doi:10.1006/jabr.1999.8035

Cited by 28 publications

(35 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the 1990's, Ulrich Oberst resumed Gröbner's tradition in Innsbruck, Tyrolia, with a sequence of important papers (specifically [Obe95] and [Obe99]) on this subject. These papers inspired the writing of this chapter.…”

Section: Linear Partial Differential Equations With Constant Coefficimentioning

confidence: 99%

See 1 more Smart Citation

Solving Systems of Polynomial Equations

Sturmfels

2002

CBMS Regional Conference Series in Mathematics

629

569

View full text Add to dashboard Cite

Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas. The set of solutions to a system of polynomial equations is an algebraic variety, the basic object of algebraic geometry. The algorithmic study of algebraic varieties is the central theme of computational algebraic geometry. Exciting recent developments in symbolic algebra and numerical software for geometric calculations have revolutionized the field, making formerly inaccessible problems tractable, and providing fertile ground for experimentation and conjecture.The first half of this book furnishes an introduction and represents a snapshot of the state of the art regarding systems of polynomial equations. Afficionados of the well-known text books by Cox, Little, and O'Shea will find familiar themes in the first five chapters: polynomials in one variable, Gröbner bases of zero-dimensional ideals, Newton polytopes and Bernstein's Theorem, multidimensional resultants, and primary decomposition.The second half of this book explores polynomial equations from a variety of novel and perhaps unexpected angles. Interdisciplinary connections are introduced, highlights of current research are discussed, and the author's hopes for future algorithms are outlined. The topics in these chapters include computation of Nash equilibria in game theory, semidefinite programming and the real Nullstellensatz, the algebraic geometry of statistical models, the piecewiselinear geometry of valuations and amoebas, and the Ehrenpreis-Palamodov theorem on linear partial differential equations with constant coefficients.Throughout the text, there are many hands-on examples and exercises, including short but complete sessions in the software systems maple, matlab, Macaulay 2, Singular, PHC, and SOStools. These examples will be particularly useful for readers with zero background in algebraic geometry or commutative algebra. Within minutes, anyone can learn how to type in polynomial equations and actually see some meaningful results on the computer screen.

show abstract

Section: Linear Partial Differential Equations With Constant Coefficimentioning

confidence: 99%

“…An algorithm for computing Noetherian operators for a given primary ideal Q was given by Oberst [Obe99]. Just like the proof in [Bjö79], Oberst's approach is based on Noether normalization.…”

Section: How To Solve Monomial Equationsmentioning

confidence: 99%

Solving Systems of Polynomial Equations

Sturmfels

2002

CBMS Regional Conference Series in Mathematics

629

569

View full text Add to dashboard Cite

show abstract

“…This is done, in rather abstract terms, in References [22,23] and, in very concrete and computational terms, in References [24,25] for the case of special ideals. The general case is still open, despite the work of many in the area.…”

Section: Algebraic Methods In Clifford Analysismentioning

confidence: 99%

Computational algebraic analysis methods in Clifford analysis

Sabadini

Struppa

2002

Math Methods in App Sciences

View full text Add to dashboard Cite

SUMMARYRecent advances in Computer Algebra have made it possible the study of algebraic analysis in an explicit and computational way. In this paper we show how these ideas have allowed the solution of a new class of problems in Cli ord analysis and we describe the computational techniques that we have to develop to this purpose. Copyright ? 2002 John Wiley & Sons, Ltd. ALGEBRAIC ANALYSISIn the papers [1][2][3][4][5][6][7][8] and also in the paper [9] in this volume, we have explicitly applied the methods of the algebraic analysis to systems of linear partial di erential equations with constant coe cients, that were widely studied by Ehrenpreis, Palamodov and by M. Sato and his collaborators. We have mainly applied this theory to systems arising in Cli ord analysis, but the procedure we will describe has general validity. For sake of completeness and to ÿx the notation, we will repeat here the basic ideas we have followed and we refer the reader to the fundamental books [10,11] and, for the applications to Cli ord analysis to the forthcoming book [12]. More details on the various cases treated here can be found in the papers mentioned above.Let R = C[z 1 ; : : : ; z n ] be the ring of polynomial with complex coe cients in the n variables z 1 ; : : : ; z n and let P = [P ij ] be an r 1 × r 0 be a matrix with entries in R. If we denote by D the vector (−i@ x1 ; : : : ; −i@ xn ), the matrix P(D) turns out to be a matrix of linear partial di erential operators with constant coe cients and the matrix equation P(D)f =0, wheref = (f 1 ; : : : ; f r0 ) is an r 0 -tuple of real di erentiable functions, represents a r 1 × r 0 system of linear partial di erential equations:

show abstract

“…Also notice that the inverse problem of the -id.int.problem consists of calculating the evaluated differential forms ϕ ∈ , knowing in advance a system of generators of the interpolating ideal . This is a particular case of a more general problem solved in [14] for primary ideals.…”

Section: 2mentioning

confidence: 97%