MinimalPolynomial[] for expressions containing symbols - wolfram-mathematica

Question

I’m using Mathematica to solve some simple equations relating to geometry, and subsequently hard-coding those solutions in a different language†. Rather than have many pages of code, it would be more concise to code the solutions as a root of a polynomial.
Let’s take:
Solve[{
dist^2 == xstep^2 + ((h - 2 r)/(NR - 1))^2,
dist^2 == (w - 2 r - NC xstep)^2 + (h/2 - r - dist/2)^2
},{xstep, dist}]
That generates a “very large output”, heavy with fractions and square roots and fourth roots. Obviously the two solved variables are the roots of quartic equations.
Please, is there a version of MinimalPolynomial[] that will work on expressions containing symbols? All that’s wanted is the five coefficients of dist’s quartic.
Thank you.
† The “different language” is PostScript, and I really don’t have the expertise to write a //PostScriptForm function. Indeed, finding the optimal balance between recomputing repeated expressions and using “… dup … roll” would, in the general case, be slow.

Answer 1

I wrote a package called "Substitutions" (archived here), which pulls out a hierarchy of sub-expressions, minimizing the coding for complex expressions. It was included in the old MathSource library. Here's the description:
It is often useful, especially when using Mathematica for software
development, to apply substitutions to complex expressions to reduce
their form. For large expressions, this task can become tedious.
Substitutions[] was designed help with the process of finding a useful
set of substitutions to simplify an expression.
It is quite old now, but should still work.

Answer 2

Reduce may be what you want:
I've consolodated some of your symbol groups into A,B,C (not nesesary, makes it fit on screen)
Reduce[{dist^2 == xstep^2 + (A)^2 &&
dist^2 == (C - NC xstep)^2 + (B - dist/2)^2 , {xstep, dist}]]
this produces a fairly large output with a bunch of conditions.
If you have known constraints that preclude various degenrate cases it helps to specify (I made these up)
$Assumptions = B != 0 && B^2 != 3 C^2 && NC^2 != 3/4;
note $Assumptions is used by Simplify, but you need to explicitly add it to the Reduce expression..
Simplify[Reduce[{dist^2 == xstep^2 + (A)^2 &&
dist^2 == (C - NC xstep)^2 + (B - dist/2)^2 && $Assumptions }, {xstep, dist}]]
output.. not too unwiledy .. The Root expression contains the coefficients you seek..
(xstep ==
Root[9 A^4 - 40 A^2 B^2 + 16 B^4 - 24 A^2 C^2 + 32 B^2 C^2 +
16 C^4 + (48 A^2 C NC - 64 B^2 C NC -
64 C^3 NC) #1 + (18 A^2 - 40 B^2 - 24 C^2 - 24 A^2 NC^2 +
32 B^2 NC^2 + 96 C^2 NC^2) #1^2 + (48 C NC -
64 C NC^3) #1^3 + (9 - 24 NC^2 + 16 NC^4) #1^4 &, 1] ||
xstep ==
Root[9 A^4 - 40 A^2 B^2 + 16 B^4 - 24 A^2 C^2 + 32 B^2 C^2 +
16 C^4 + (48 A^2 C NC - 64 B^2 C NC -
64 C^3 NC) #1 + (18 A^2 - 40 B^2 - 24 C^2 - 24 A^2 NC^2 +
32 B^2 NC^2 + 96 C^2 NC^2) #1^2 + (48 C NC -
64 C NC^3) #1^3 + (9 - 24 NC^2 + 16 NC^4) #1^4 &, 2] ||
xstep ==
Root[9 A^4 - 40 A^2 B^2 + 16 B^4 - 24 A^2 C^2 + 32 B^2 C^2 +
16 C^4 + (48 A^2 C NC - 64 B^2 C NC -
64 C^3 NC) #1 + (18 A^2 - 40 B^2 - 24 C^2 - 24 A^2 NC^2 +
32 B^2 NC^2 + 96 C^2 NC^2) #1^2 + (48 C NC -
64 C NC^3) #1^3 + (9 - 24 NC^2 + 16 NC^4) #1^4 &, 3] ||
xstep ==
Root[9 A^4 - 40 A^2 B^2 + 16 B^4 - 24 A^2 C^2 + 32 B^2 C^2 +
16 C^4 + (48 A^2 C NC - 64 B^2 C NC -
64 C^3 NC) #1 + (18 A^2 - 40 B^2 - 24 C^2 - 24 A^2 NC^2 +
32 B^2 NC^2 + 96 C^2 NC^2) #1^2 + (48 C NC -
64 C NC^3) #1^3 + (9 - 24 NC^2 + 16 NC^4) #1^4 &, 4]) &&
3 A^2 + 4 B dist + xstep (8 C NC + 3 xstep) ==
4 (B^2 + C^2 + NC^2 xstep^2)