5 ms·

I'm a huge fan of SAGE! I keep using it to discover new math. I generally play with the associate Sage module of any algebraic structure to get a real feel for

by bollu 6y ago

I'm a huge fan of SAGE! I keep using it to discover new math. I generally play with the associate Sage module of any algebraic structure to get a real feel for what is going on. Some examples of: 1. Grobner basis: http://bollu.github.io/#computing-equivalent-gate-sets-using-grobner-bases http://bollu.github.io/#computing-equivalent-gate-sets-using... 2. Commutators of subgroups: http://bollu.github.io/#5age-code-to-play-around-with-commutators-of- http://bollu.github.io/#5age-code-to-play-around-with-commut... 3. Graph/group theory: https://github.com/bollu/bollu.github.io/blob/2f88c28efc54fb16463e2d9e0ac63f28620e223b/content/blog/math/intro_to_sage.md https://github.com/bollu/bollu.github.io/blob/2f88c28efc54fb...

Alkhwarizmi 6y ago
I LOVE your blog man! Keep it up
abdullahkhalids 6y ago
Completely off topic question. Can Grobner basis techniques be used to solve a set of quadratic inequalities?
- thechao 6y ago
  Not ... really? I usually compare the Gröbner basis to generalized matrix decomposition in the sense of solving systems of linear equations. If you’re solving systems of quadratic equations, you should look into interior barrier solving methods, which generalize the simplex method.
  - blt 6y ago
    Really? I thought the simplex method was kind of the opposite of an interior point method.
- generationP 6y ago
  Gröbner bases appear in the standard reference on real algebraic geometry ( https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted3.pdf https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-po... ), so the answer may well be "yes", but don't ask me for the details. (All I can tell is that quadratic systems are no easier than degree-n systems in general -- there's a known trick how any system can be reduced to degree 2 by introduced extra variables.)
  - ycreader 6y ago
    Can you tell me the name of the trick to read about it on wikipedia? thanks
- enriquto 6y ago
  > Can Grobner basis techniques be used to solve a set of quadratic inequalities? One application of Gröbner bases is to solve systems of polynomial equations [0], and characterize their solutions. In many circumstances it is not however the best method, for example if you are content with a numerical solution you can use other techniques. Notice that this concerns equations, not inequalities. After a cursory search, I cannot find any "big" reference for solving polynomial inequalities using Gröbner bases. There is this [1], still, not sure how serious it is. [0] https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis#Solutions_of_a_system_of_algebraic_equations https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis#Solutions_o... [1] https://arxiv.org/abs/1603.01183 https://arxiv.org/abs/1603.01183

by bollu 6y ago

Alkhwarizmi 6y ago

I LOVE your blog man! Keep it up

abdullahkhalids 6y ago

Completely off topic question. Can Grobner basis techniques be used to solve a set of quadratic inequalities?

thechao 6y ago
Not ... really? I usually compare the Gröbner basis to generalized matrix decomposition in the sense of solving systems of linear equations. If you’re solving systems of quadratic equations, you should look into interior barrier solving methods, which generalize the simplex method.
- blt 6y ago
  Really? I thought the simplex method was kind of the opposite of an interior point method.
generationP 6y ago
Gröbner bases appear in the standard reference on real algebraic geometry ( https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted3.pdf https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-po... ), so the answer may well be "yes", but don't ask me for the details. (All I can tell is that quadratic systems are no easier than degree-n systems in general -- there's a known trick how any system can be reduced to degree 2 by introduced extra variables.)
- ycreader 6y ago
  Can you tell me the name of the trick to read about it on wikipedia? thanks
enriquto 6y ago
> Can Grobner basis techniques be used to solve a set of quadratic inequalities? One application of Gröbner bases is to solve systems of polynomial equations [0], and characterize their solutions. In many circumstances it is not however the best method, for example if you are content with a numerical solution you can use other techniques. Notice that this concerns equations, not inequalities. After a cursory search, I cannot find any "big" reference for solving polynomial inequalities using Gröbner bases. There is this [1], still, not sure how serious it is. [0] https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis#Solutions_of_a_system_of_algebraic_equations https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis#Solutions_o... [1] https://arxiv.org/abs/1603.01183 https://arxiv.org/abs/1603.01183