August 10th, 2018, 11:31 AM | #11 |
Math Team Joined: May 2013 From: The Astral plane Posts: 1,889 Thanks: 768 Math Focus: Wibbly wobbly timey-wimey stuff. | |
August 10th, 2018, 01:22 PM | #12 |
Senior Member Joined: Oct 2009 Posts: 551 Thanks: 174 |
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 01:27 PM. |
August 10th, 2018, 06:37 PM | #13 | |
Senior Member Joined: May 2016 From: USA Posts: 1,148 Thanks: 479 | Quote:
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. | |
August 10th, 2018, 09:15 PM | #14 | |
Senior Member Joined: Sep 2016 From: USA Posts: 471 Thanks: 262 Math Focus: Dynamical systems, analytic function theory, numerics | Quote:
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. | |