What would be the best MacBook Pro for the following kind of calculations, an M1 Max w/ 64GB RAM or a maxed out Intel (something like 2.4 Ghz 8-core i9, 64GB RAM)?
Here's an example with small numbers, namely 3 and 5. The goal is to determine the minimal polynomials of certain root sums. Consider the polynomials
x ^ 3 - x - 1 and x ^ 5 - x - 1 .
We find their largest real roots by using
x / . NSolve [ y==0 , x , WorkingPrecision->90 ] ,
first with y the first above polynomial, and then with y the the second one. We then add these two roots. (They are 1.324...... and 1.167...... with sum 2.492........ .)
To get the minimal polynomial of their sum (call it s ) use
(MinimalPolynomial [ RootApproximant [ s ] ] / . # 1 - > x ) [[1]] .
In this case we find that the probable minimal polynomial is
-19 + 19 x + 12 x ^ 2 + ,,,, - 5 x ^ 12 - 5 x ^ 13 + x ^ 15
In fact, it is the minimal polynomial.
I would want to do calculations of more or less this type with numbers larger than 3 and 5.
This simple direct approach will not work on any computer once the numbers are too big,
but what available MacBook Pro would allow me to go as far as possible?