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Author here. Gröbner bases are powerful tools for us to understand polynomials. I started on this article when I encountered them during research, and decided t
by ransac9000 3y ago
Author here. Gröbner bases are powerful tools for us to understand polynomials. I started on this article when I encountered them during research, and decided to write down some notes. After a few months, these notes turned into this blog post. It's quite a long read, and let me know if you have any questions.
- vasvir 3y agothanks for writing this. I have read about 1/3 to 1/2 so far and it is very nicely written and easy to follow.
- ransac9000 3y agoNo problem, and thank you for the kind words!
- abdullahkhalids 3y agoCan the method of Grobner bases also be used to solve systems of polynomial inequalities?
- ransac9000 3y agoIt seems that you can reformulate inequalities into equalities and solve the system: https://arxiv.org/pdf/1603.01183.pdf https://arxiv.org/pdf/1603.01183.pdf
- nsajko 3y agoInequalities can be transformed into equations quite easily, by adding variables: Consider a strict inequality with real variables x_i: f(x_1,...,x_n) > 0 By adding the variable y we obtain an equivalent equation: f(x_1,...,x_n) = y^-2 This is correct because y^-2 is always strictly positive. If f is a polynomial, the above can be written as a polynomial equation like so: f(x_1,...,x_n) * y^2 = 1 To transform a non-strict inequality into an equation, on the other hand, the procedure is the same, just use 2 instead of -2 as the exponent. That is, this inequality: f(x_1,...,x_n) ≥ 0 ... is equivalent to this equation: f(x_1,...,x_n) = y^2
- enedil 3y agoTrouble starts when your field doesn't always have a solution for f(x_1,...,x_n) = y^2, like rational numbers (but I'm sure you can find more examples). But maybe that can be mitigated as well?
- JadeNB 3y agoI'm not an expert in this area, but, precisely for the reason you mention, I'd expect it's easier to solve a polynomial inequality over ℚ by solving it over ℝ and intersecting down to ℚ, rather than by working directly over ℚ.
- macrolocal 3y agoAs long as you don’t bump into Gal(ℝ/ℚ) somewhere along the way.
- cevi 3y agoFor systems of polynomial inequalities, the appropriate tool is Cylindrical Algebraic Decomposition (this tool generalizes to systems of exponential inequalities as well).
- jjtheblunt 3y agonicely done
- ransac9000 3y agothanks!
- red_trumpet 3y agoLooks nice! A tip for your latex: I would use \langle and \rangle to surround the generators of an ideal, which looks better than using < and >. And I think they are called angle brackets, not square brackets (which would be [,]).
- ransac9000 3y agoGood catch! I thought I changed all < and > into \langle and \rangle.
- bainsfather 3y agoA bit after halfway, following: "In this case, .. , and we have", I think you need a "-" sign rather than a "+" in your equation?
- ransac9000 3y agoGood catch! Thanks for the feedback.