Determining the number of independent equations

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In summary, the conversation involves the difficulty of finding a minimal system of generating polynomials for a given ideal in geometry, which ultimately leads to the concept of Gröbner bases in algebraic geometry. The problem is further complicated by the non-linearity of the polynomials and the dependence of variables. There is no known general solution to this problem, although certain cases can be solved using basic geometry and algebra rules.
  • #1
guideonl
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geometry, determining number of independet equations
Hi everyone,
In geometry problems involved with triangles (inside a circle in my case), I can write different equations based on some known measures using pythagoras theorem, similarity , trigonometric and geometric relations. But, if I write all these equations I propably get some dependent equations.
My question is: how can I avoide depending equations? in other words, for n given triangles, how many independet equations/relations i can write?
Also, please note if there are dependent equations/relations in which using one- you may not use the other, otherwise you get dependent equations.

Thanks
 
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  • #2
The important part first: I do not know an answer.

In geometry we have usually linear and quadratic equations, that is a set of polynomials ##S=\{\,p_1,\ldots,p_n\,\}## which generate an ideal ##\mathcal{I} \subseteq \mathbb{R}[X_1,\ldots,X_m]##.
Now there are some problems: the ##p_i## are not linear, the variables are not independent, i.e. we have uniqueness up to ##\mathcal{I}## and we do not know ##\mathcal{I}##, and there are many variables: angles and distances. A geometric statement is thus the question whether a certain polynomial is in ##\mathcal{I} ## or not.

I guess the main problem will be ##\mathcal{I} ## and your question is: How can we find a minimal system of generating polynomials for ##\mathcal{I} ?## This leads via algebraic geometry into ring theory. However, there is no general way to determine the generators of ##\mathcal{I} ## that I knew of, other than in the case of linear polynomials.
 
  • #3
Thank you fresh_42...
Well, you took the problem far away.. I meant using basic geometry & algebra rules...
 
  • #4
guideonl said:
Thank you fresh_42...
Well, you took the problem far away.. I meant using basic geometry & algebra rules...
Your thread is labeled with an "A" prefix, meaning that you want the discussion at the Advanced / graduate school level. I'll change it for you now to "I" = Intermediate (undergraduate level). :smile:
 
  • #5
guideonl said:
Thank you fresh_42...
Well, you took the problem far away.. I meant using basic geometry & algebra rules...
I ran so many times in complicated circles, arriving with an equation I could have started with, that I doubt there is an easy solution. You can only list all variables and attach to every variable what you know, e.g. given a triangle intersected by one of its heights, then we have ##\alpha+\beta+\gamma =\pi## but may only add one of the partial triangles to the list of equation. However, we might need an angle in the other part and all of a sudden we have a redundancy.
 
  • #6
guideonl said:
Thank you fresh_42...
Well, you took the problem far away.. I meant using basic geometry & algebra rules...
Here is the corresponding framework: Gröbner bases.
 

Related to Determining the number of independent equations

What is the purpose of determining the number of independent equations?

The purpose of determining the number of independent equations is to understand the relationship between the variables in a system and to determine the minimum number of equations needed to solve for all the unknown variables.

How do you determine the number of independent equations?

The number of independent equations can be determined by counting the number of unique variables in the system and subtracting the number of constraints or known relationships between those variables.

What is the significance of having the correct number of independent equations?

Having the correct number of independent equations is crucial in solving a system of equations as it ensures that there are enough equations to solve for all the unknown variables. If there are too few equations, the system may have multiple solutions, and if there are too many equations, the system may be overdetermined and have no solution.

Can there be more than one solution to a system of equations with the same number of independent equations?

Yes, it is possible for a system of equations to have multiple solutions even with the correct number of independent equations. This can occur when there are redundant or dependent equations in the system.

How does the number of independent equations affect the complexity of solving a system?

The number of independent equations directly affects the complexity of solving a system. Generally, the more independent equations there are, the simpler the system is to solve. However, if there are too many equations, the system may become computationally complex to solve.

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