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Variations on Bezout's theorem

General Information

  • Lecture: 2h, 5 LP
  • official name: "Modul Algebra F5: Fortgeschrittene Themen der Algebra"

This course can serve as a preparation for the anticipated Thematic Einstein Semester Varieties, Polyhedra, Computation during the winter term 2019/20.

News

  • The table of contents is now available.
  • The last lecture will be on 4th July.
  • There will be NO lecture on 20th June. The REPLACEMENT lecture is on Wednesday, 26th June at 16:15 in room MA 316. (The lecture on 27th June takes place as usual.)
  • There will be NO lecture on 6th June. The REPLACEMENT lecture is on Wednesday, 5th June at 16:15 in room MA 316.
  • There will be NO lecture on 30th May due to holiday (Christi Himmelfahrt).

Schedule

Course Hours
Type
Day
Hours
Room
Lecturer
Lecture
Thursday
1615-1745
MA 316
Prof. Dr. Peter Bürgisser

Summary and Table of Contents

Bezout's Theorem is a fundamental result of algebraic geometry. There are important extensions to polynomial system with few terms (fewnomials) that build a bridge to convex geometry (toric varieties, Newton polytopes, mixed volume). The goal of the lecture is to prove and discuss these results from different angles. We shall mainly work over the complex numbers and rely also on tools from differential geometry or integral geometry, that will be devoloped in the lecture.

The table of contents is available under this link.

Literature

  • Peter Bürgisser and Felipe Cucker. Condition: The Geometry of Numerical Algorithms. Springer 2013.
  • Gregorio Malajovich. Nonlinear Equations. Impa 2011.
    Available as pdf under the following link
  • Frank Sottile. Real Solutions to Equations From Geometry. University Lecture Series volume 57, AMS, 2011.

The book by Sottile is available as e-book. It has been licensed by the TU Berlin last year. Unfortunately, this information is not yet visible in Primo. Here is the legal link to the e-book: http://www.ams.org/books/ulect/057/ulect057.pdf
There is also an arXiv version with ID number 0609829.

Semesterapparat: We created for each of the lectures "Condition" and "Variations on Bezout's Theorem" a so called Semesterapparat. This ensures that some of the proposed books are always present for study in the mathematical library.

Zusatzinformationen / Extras

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Auxiliary Functions