0

If a system of polynomial equations has infinitely many solutions over an algebraically closed field $\mathbb{C}$, what is the dimension of the solution set? How to analyze this question in general?

I am curious about this question, thus I found this tutorial, however it seems stops at the question $(a)i$, and does not give the answer of $(a)ii$, and $(b)$.

Could someone give a hint on this? Thanks a lot!

links to the tutorial: https://www.math.usm.edu/perry/old_classes/mat681sp14/gbasis_notes.pdf enter image description here

----------An update on the concept of dimension:----------

If there are infinitely many solutions to the polynomial system, the number of solutions does not make sense for counting solutions, instead, we need something else: dimension. In linear algebra, if the solutions to a linear system are infinitely many, then we care about how many basis are there to form the solution space, and the number of basis is called dimension of the solution space (am I right?). So, in polynomial systems, I am curious about the similar question, how to define such dimension?

In addition, I have read the chapter on structure of varieties, and found that:

  1. The solution set of polynomial system is defined (called) as algebraic variety

  2. Thus the dimension of solution set is the dimension of algebraic variety

  3. The dimension of algebraic variety $X$ is defined as the length of longest decreasing chain of irreducible sub-varieties of $X$ (refers to definition 3.2.8)

I am wondering whether this definition of dimension of variety is the thing that is similar to the example of linear algebra?

Moreover, regarding the dimension of solution set, how many possible definitions there exists?

(I am a very new beginner of algebraic geometry, and my intension is to understand how people characterize the solution set of polynomial equations, so that I can understand how to characterize solutions to the system in my hand.)

TShiong
  • 1,257
M.K
  • 541
  • 1
    The solution set is not a vector space in general. So you need to say what the "dimension" of such a set is. See for example for the Gelfand-Kirillov dimension. The Groebner basis already answers some questions. See the chapter on it in your text. – Dietrich Burde Feb 24 '22 at 14:40
  • @DietrichBurde Ah I also read the website, but just the commutative case. So if the polynomials are over a field (so it is a commutative ring), then we use Krull dimension for defining the dimension of solution space. Am I right? – M.K Feb 24 '22 at 15:01
  • Solution set will be an (affine) variety. – Wuestenfux Feb 24 '22 at 15:40
  • 1
    It's a very broad topic to describe the "dimension" of solutions to systems of polynomial equations in several variables (as suggested by your image). I'm voting to close for lack of focus (too broad), but I'd be agreeable to reopen if the Question was narrowed. – hardmath Mar 04 '22 at 16:33
  • In the way of suggesting a narrowing of the Question, why not focus on the notion of dimension of solutions sets, if your impression is that the tutorial raised that issue but then abandoned it? Please include thoughts such as "then we use Krull dimension," hopefully with some explanation, in the body of the Question itself (not merely in the Comments). It is in present form quite broad, but it might help many Readers if one such aspect of your interest was given a detailed exposition. – hardmath Mar 07 '22 at 18:03
  • @hardmath Thank you for your suggestion! I just update my question and hopefully this time it is more clear. – M.K Mar 07 '22 at 20:59
  • @hardmath this is not a broad topic, this is very specifically described in the (introductory) algebraic geometry literature. Every first course text contains a section or chapter about it. There are many questions on MSE dealing with this - for instance this shows that dimension of a variety recovers the linear-algebra dimension of a vector space when considering a vector space as a variety. – KReiser Mar 07 '22 at 22:39
  • The original question is a duplicate of https://math.stackexchange.com/questions/2298162/algebraic-variety-of-dimension-0 and/or https://math.stackexchange.com/questions/2634409/can-a-variety-of-dimension-geqslant-1-be-finite and possibly others, but it would be nice if the OP could clarify exactly what they're after here. – KReiser Mar 07 '22 at 22:41
  • @hardmath That doesn't really change anything. "How do we define dimension in algebraic geometry" is a fairly specific question with a good, focused answer. In the future, before voting to close a question as needing more focus, please make sure that's really the most pressing issue with the problem. (It's very likely a duplicate, it could be closed for lack of research effort, etc etc etc, but it clearly does not meet the criteria of being too broad.) – KReiser Mar 09 '22 at 04:42
  • @KReiser: You are welcome to ping me in chat if you wish to discuss this. – hardmath Mar 09 '22 at 06:29

0 Answers0