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August 10th, 2018, 12:31 PM   #11
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Originally Posted by SDK View Post
These problems are toxic.
I agree completely.

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August 10th, 2018, 02:22 PM   #12
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Some Grobner base theory easily finds an equivalent system:
$$c^4-44c^2+448=0$$
$$24b + c^3-28c=0$$
$$14a+b^2c +bc^2-21b+c^3-21c=0$$
Solve these and you'll get the solution to the equivalent system.

This method ALWAYS works with solving systems of polynomial equations.

Last edited by Micrm@ss; August 10th, 2018 at 02:27 PM.
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August 10th, 2018, 07:37 PM   #13
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Nope I was in a hurry and made a mistake. I thought the sum had a factor of $a$. I strongly dislike questions like this at a lower level since it gives the impression that math is just a bunch of clever tricks to be remembered.

In reality, despite the large amount of structure in this problem, it is still extremely difficult or impossible to solve most nonlinear systems exactly. Even in the case you know ahead of time that there are integer solutions, trying to eliminate variables is still the worst possible way to proceed. In this case you would just do Newton and round to the nearest integer.

There is no case where playing around with the equations is the "right" thing to do yet at low level this is the only thing students are equipped to do. These problems are toxic.
I cannot agree in the abstract that the question itself is toxic.

For example, if the context in which the question was given was to teach that systems of quadratic or cubic or even quartic equations in n unknowns have exact solutions derivable by formula, but that the process can be very burdensome using such basic tools, this problem is passable. It is even then not a good question because the intended point can be evaded due to skipjack's elegant solution.

In practice, I am reasonably confident that the problem was not posed in such a context, but that it was designed to show a trick of very limited applicability. It is that which is toxic.
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August 10th, 2018, 10:15 PM   #14
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Originally Posted by Micrm@ss View Post
Some Grobner base theory easily finds an equivalent system:
$$c^4-44c^2+448=0$$
$$24b + c^3-28c=0$$
$$14a+b^2c +bc^2-21b+c^3-21c=0$$
Solve these and you'll get the solution to the equivalent system.

This method ALWAYS works with solving systems of polynomial equations.
This isn't quite right.

First off, the Groebner basis algorithm has double exponential running time in both CPU and memory. This puts extremely tight restrictions on what can be practically computed.

Secondly, Groebner bases give an easy proof that a system of polynomial equations has no solution. They do not help at all AFAIK when a system has infinitely many solutions.

In the case that a system has only finitely many solutions, they provide a method (in theory) for finding solutions. However, this does not give exact solutions and is often not of any practical use due to nonuniqueness of multivariate pseudoremainders and the fact that pseudoremainders are numerically unstable.

In practice, one would use the cylindrical algebraic decomposition for this problem (assuming the coefficients are algebraic). Note that it is also has double exponential runtime so it does not avoid the first issue.
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