How to formulate this problem in Mathematica?

Question

Say, I have the following expressions

f = R[p]
g = S[p]

where R,S are rational functions, and I want to find a polynomial such that

P[f,g]=C(onstant)

For example:

f=p^2 + p
g=p+4

leading to

P(x,y)=y^2-x-7y
g^2-f-7g=-12.

Can Mathematica help me in any way finding the polynomial P, if R,S become more and more complex ( but still rational, of course )?

And, at least, how can I find the answer to the example with Mathematica?

Answer 1

Start by making a list of polynomial terms to choose from: for example, all terms up to third order in $f$ and $g$,

terms = DeleteDuplicates[Times @@@ Tuples[{1, F, G}, 3]]
(*    {1, F, G, F^2, F G, G^2, F^3, F^2 G, F G^2, G^3}    *)

Express these terms as polynomials in $p$, and list their polynomial coefficients (here up to sixth order in $p$):

f = p^2 + p;
g = p + 4;
X = CoefficientList[terms /. {F -> f, G -> g}, p, 7]
(*    {{ 1,  0,  0,  0,  0,  0,  0},
       { 0,  1,  1,  0,  0,  0,  0},
       { 4,  1,  0,  0,  0,  0,  0},
       { 0,  0,  1,  2,  1,  0,  0},
       { 0,  4,  5,  1,  0,  0,  0},
       {16,  8,  1,  0,  0,  0,  0},
       { 0,  0,  0,  1,  3,  3,  1},
       { 0,  0,  4,  9,  6,  1,  0},
       { 0, 16, 24,  9,  1,  0,  0},
       {64, 48, 12,  1,  0,  0,  0}}    *)

Find the null-space of this matrix: the polynomials that combine to zero,

NullSpace[Transpose[X]] . terms
(*    {84 - 7 F - 37 G - F G + G^3,
       12 F - F^2 - 7 F G + F G^2,
       12 - F - 7 G + G^2}             *)

Verify these three solutions:

% /. {F -> f, G -> g} // Expand
(*    {0, 0, 0}    *)

So we have the solutions $$ -f g-7 f+g^3-37 g+84 = 0\\ -f^2+f g^2-7 f g+12 f = 0\\ -f+g^2-7 g+12 = 0 $$

There will be many more solutions if you increase the order of terms in terms. Factoring these solutions will show you the minimal function $P$, which in this case is $P(f,g)=-f+g^2-7 g+12=0$. You can add constants $C$ to this solution at will.

Rational functions

An extension of the above procedure to rational functions is straightforward through extending all functions by the common denominator. As an example,

f = p/(p + 1);
g = p*(p - 1)/(p^3 + 7);
denom = PolynomialLCM @@ Denominator /@ {f, g}
(*    (1 + p) (7 + p^3)    *)

Expanding the rational functions by the denom gives polynomials of order

e = Max@Exponent[{f, g}*denom, p]
(*    4    *)

Use polynomial terms up to fifth order in the result:

maxorder = 5;
terms = DeleteDuplicates[Times @@@ Tuples[{1, F, G}, maxorder]]
(*    {1, F, G, F^2, F G, G^2, F^3, F^2 G, F G^2, G^3, F^4, F^3 G,
       F^2 G^2, F G^3, G^4, F^5, F^4 G, F^3 G^2, F^2 G^3, F G^4, G^5}    *)

Compute the coefficients matrix:

X = CoefficientList[terms*denom^maxorder /. {F -> f, G -> g}, p, e*maxorder + 1]
(*    21x21 matrix with integer coefficients    *)

Pick the null-space solution that has the smallest number of non-zero coefficients:

MinimalBy[NullSpace[Transpose[X]], Total@*Unitize] . terms
(*    {-F + 3 F^2 - 2 F^3 - 7 G + 21 F G - 21 F^2 G + 6 F^3 G}    *)

So we see that $$ 6 f^3 g-21 f^2 g-2 f^3+3 f^2+21 f g-f-7 g = 0 $$