The lecture focuses on finding combinatorial information from the quotient
a a polynomial ring by an ideal generated by powers of linear forms. With
powers of sums of the variables the dimension of the quotient gives the
number of forests and the number of trees on the number of variables
vertices. The Hilbert series of these algebras is the Hilbert series of
the algebra of powers of standard monomials where the powers are given by
parking functions. Given a hyperplane arrangement the ideal generated by
all the one dimensional intersections of hyperplanes taken to the power
that is the number of hyperplanes that intersect to give each line. It
turns out that the dimension of the quotient algebra gives the number of
bases and the number of independent sets in the matroid of the
arrangement.

Keywords:

Hyperplane arrangements; Motkin numbers; parking functions; pseudo-Grobner basis.

Lecture #10724

Need help? Visit our help pages at http://www.msri.org/communications/vmath/hints

Supplements

Right click on the link and "Save As..." to save to your local computer.

• 10724-Postnikov.pdf (0.2 MB)

Left click on thumbnail to see a larger image.

Streaming Video

This is a high quality streaming video encoded with MPEG-4 and with 640x480 resolution.

Windows and Mac users, QuickTime 6.5 or later required
Linux users, please see our Linux Help Page on how to view our streaming videos

Follow this link to --- Watch the Video Now Via Streaming Video ---

Download QuickTime File

You can download the QuickTime file here. Right click on the link and "Save As..." to save to your local computer.

• 10724-10724-QuickTime.mov (437 MB)

Download MPEG File

You can download the MPEG file here. Right click on the link and "Save As..." to save to your local computer.

• 10724-10724-MPEG-1 DVD.mpg (883 MB)

Create a DVD

You can download the video and audio files here. Please note that you need both files to create a DVD. Right click on the link and "Save As..." to save to your local computer. You can find instructions on how to create a DVD on our help page at http://www.msri.org/communications/vmath/author

• 10724-10724-DVD PCM Audio.aiff (646 MB - Audio Only)
• 10724-10724-MPEG-2 120min High Quality Encode.m2v (1472 MB - Video Only)

Buy the DVD

If none of the options work for you, you can always buy the DVD of this lecture.

If you would like to purchase a copy of this video for $15+shipping, please Click Here!

See more of our Streaming Videos on our main VMath - Streaming Video page.