Talk:Toric ideal

Trying to understand the definition

Parsing the only text in the article:

... a toric ideal is an ideal generated by differences of two monomials. An affine or projective algebraic variety defined by a toric prime ideal or a homogeneous toric ideal is an affine or projective toric variety.

I am hereby trying to figure out what does that mean by using my knowledge of the terms at wikilinks, but ignoring the literature on the toric ideals themselves, expecting the encyclopedia article to guide me. I am a perfect man on the Clapham omnibus to try it, as I happen to know the basic algebra, but at this point (really!) having no clue about the toric ideals.

Substituting the definitions found through the links:

ideal is a smart word for an (sub)algebraic ring formed by multiplication of an element of a bigger ring by a counter (n*A, a.k.a. repeated additions A+A+A..., think integers divisible by A, like even for A=2: 1*2, 2*2, 3*2, etc. prime ideal is when A is kinda prime, think the same example with A=2 , 3, or 5, but not 4)
difference is a smart word for subtraction (-)
monomial is a smart word for a term of a polynomial, i.e. $x,y,x\cdot y,3\cdot y$ , etc. Are (1) the degrees of monomials fixed? ... (2) are the monomials primitive (i.e., without a constant)? I assume that in order to form an interesting (additive) ring the answer to #1 is "yes", and to #2 is "no" ... (3) wait, are the constants in monomials integer? Based on prime properties (below) I think (but not sure) the answer to #3 is also "yes".
we end up with a construct $n(ax-by)$ , where n, a, b are all integers. x and y can obviously replaced by some products of powers of x and y, but I am trying to figure out the simplest case.

@D.Lazard: Before I continue, can I check at least that I got it so far properly? Is the second (unparsed yet) sentence also a part of the definition or is it a property tying together few objects? Викидим (talk) 02:06, 5 January 2025 (UTC)[reply]

This article is a stub. This means that it is waiting to be expanded by specialists, and it is here as a placeholder for a future article and for people who follow a link from another article. So, it is not surprisingly that you have difficulties to understand the article. For the moment, I have not the time nor the competence for expanding the article, and answering to your questions is almost the same as expanding the article. So, my suggestion is to look at the references. As I have not checked them, I am not sure that toric ideals are defined in all of them. D.Lazard (talk) 10:08, 5 January 2025 (UTC)[reply]

Thank you. I will try. Викидим (talk) 22:40, 5 January 2025 (UTC)[reply]