[ransac9000]

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.

[red_trumpet]

Looks 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]

Good catch! I thought I changed all < and > into \langle and \rangle.

[bainsfather]

A bit after halfway, following: "In this case, .. , and we have", I think you need a "-" sign rather than a "+" in your equation?

[ransac9000]

Good catch! Thanks for the feedback.

[abdullahkhalids]

Can the method of Grobner bases also be used to solve systems of polynomial inequalities?

[nsajko]

Inequalities 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]

Trouble 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]

I'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]

As long as you don’t bump into Gal(ℝ/ℚ) somewhere along the way.

[ransac9000]

It seems that you can reformulate inequalities into equalities and solve the system: https://arxiv.org/pdf/1603.01183.pdf

[cevi]

For systems of polynomial inequalities, the appropriate tool is Cylindrical Algebraic Decomposition (this tool generalizes to systems of exponential inequalities as well).

[vasvir]

thanks 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]

No problem, and thank you for the kind words!

[jjtheblunt]

nicely done

[ransac9000]

thanks!