Finding global minima of nth degree polynomials

[Mayhem]

Is it possible (read: reasonably easy) to find global minima of an nth degree polynomial of the general form $$a_nx^n + a_{n-1}x^{n-1} ... a_2x^2 +a_1x + a_0 = 0$$ It seems to have applications in computational chemistry as I have a "hunch" that polynomial regression could be used to somewhat accurately predict the lowest conformational energy levels of complicated molecules.

Inspiration for this hunch:

Finding extrema for a given polynomial requires finding zeroes of its first derivatives. I heard this gets difficult quickly.

Any input? This is purely out of curiosity.

 

[Stephen Tashi]

Is it possible (read: reasonably easy) to find global minima of an nth degree polynomial of the general form $$a_nx^n + a_{n-1}x^{n-1} ... a_2x^2 +a_1x + a_0 = 0$$ It seems to have applications in computational chemistry as I have a "hunch" that polynomial regression could be used to somewhat accurately predict the lowest conformational energy levels of complicated molecules.

Of course, some polynomials like $f(x) = x^3$ and $f(x) = -2x^2$ do not have global minima. From you graph, it looks like local minima are what is useful.

As far as finding approximate solutions for the extrema of polynomials using computers, this is well known territory.

As far as writing an answer as a one-line formula in symbols using common mathematical functions, this is not easy and not even possible using what most people regard as common mathematical functions.

 

[Svein]

If you restrict your polynomials to a compact subset of ℝ (closed and bounded), global extrema can be found either among the local extrema or among the values on the border of the subset.