Applied Math and Media

Growing up during the Cold War in Kentucky gave my world-view a little less than complete treatment as to who I would become, what I would do for a living and how my relationships would later serve mainly to break my heart and hang me out to dry. In spite of all that, I managed to survive to my current age of 61 and am determined not to let the remainder of my life on Earth get dominated by the Evil One. I have wasted plenty of breath trying to convince certain close relatives and distant friends that something positive is going on in the world, completely separate and apart from what is reported on TV News programs. (There’s a reason they are called ‘programs’ by the way.)

Recently, I was introduced to an awesome term which to me represents that “lost chord” that The Moody Blues sent me in search of about four decades ago. If you’ve been following me along in my various web-spaces and spheres of influence (and confluence), you’ll “get”, as I have, the full scope of what this little term ‘syzygy‘ means in all of its glory as a generalized principle much akin to the meta-term SYNERGY of the 20th Century, which I learned of through reading books by R. Buckminster Fuller. Just have a look:

Syzygy (mathematics)

From Wikipedia, the free encyclopedia
For other uses, see Syzygy (disambiguation).

In mathematics, a syzygy (from Greek συζυγία ‘pair’) is a relation between the generators of a module M. The set of all such relations is called the “first syzygy module of M“. A relation between generators of the first syzygy module is called a “second syzygy” of M, and the set of all such relations is called the “second syzygy module of M“. Continuing in this way, we derive the nth syzygy module of M by taking the set of all relations between generators of the (n − 1)th syzygy module of M. If M is finitely generated over a polynomial ring over a field, this process terminates after a finite number of steps; i.e., eventually there will be no more syzygies (see Hilbert’s syzygy theorem). The syzygy modules of M are not unique, for they depend on the choice of generators at each step.

The sequence of the successive syzygy modules of a module M is the sequence of the successive images (or kernels) in a free resolution of this module.

Buchberger’s algorithm for computing Gröbner bases allows the computation of the first syzygy module: The reduction to zero of the S-polynomial of a pair of polynomials in a Gröbner basis provides a syzygy, and these syzygies generate the first module of syzygies.

Further reading

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