> 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_o...

[1] https://arxiv.org/abs/1603.01183

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.

Really? I thought the simplex method was kind of the opposite of an interior point method.

Gröbner bases appear in the standard reference on real algebraic geometry ( 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.)

Can you tell me the name of the trick to read about it on wikipedia? thanks