# How to solve a linear equation within a field of rational functions?

I have a field of rational functions over the rational numbers in 4 variables:

K = Frac(PolynomialRing(QQ,'x,y,z,w'))

and some explicit elements R,...,R[n] in K. Is there a way to solve the equation

aR + ... + a[n]R[n] == 0

with a[i] in QQ? That is, I would like to first check if there are any solutions. In the positive case, then I would like an explicit solution.

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You may evaluate the list L of rational functions at various random points to get a matrix of at most nrows:

sage: def to_matrix(L, nrows):
....:     rows = []
....:     for _ in range(nrows):
....:         (a,b,c,d) = (QQ^4).random_element()
....:         if all(p.denominator().subs(x=a,y=b,z=c,w=d)!=0 for p in L):
....:             rows.append([p.subs(x=a,y=b,z=c,w=d) for p in L])
....:     return matrix(rows)


and then, if you have a list L of let's say 5 rational functions,

sage: K = Frac(PolynomialRing(QQ,'x,y,z,w'))
sage: L = [K.random_element() for _ in range(5)]
sage: L
[(-5*x*z - 2*y*z - 1/4*w^2 + z - 2)/(y*z + 1/5*z^2 - w^2 + 1/9*x - 4/95*y),
(97/12*y*z - z^2 + 7*y + 1/114*w + 1/2)/(2*x*y - x*z + 4/5*y*z + 2*x*w + 3*z),
(4/3*y^2 - 2*z^2 + 3*x*w - 19/12*z)/(-1/5*y*z + 28/3*y*w + z*w - 4*x + 2*z),
(-x*y - 5/2*x*z + 1/4*z + 1/12)/(1/9*x*y - 4*z^2 - 137/2*z + w),
(x*y + 1/4*x*z + 1/14*y*z - 1/2*y*w + 33*w^2)/(x*w + 1/14*z)]


You may evaluate each of the rational function at let's say m>5 different random points, giving you a m x 5 matrix. If m=8, it may be:

sage: M = to_matrix(L, 8)
sage: M
[  -403845/186844     136275/60268       78217/2968         -603/107         18217/83]
[      28215/5516        -1709/228            -17/4           -39/22               15]
[  2465820/159503      37355/21804 6298795/15641408   124689/1572500         2553/361]
[        -725/124        -341/2052           -43/26           -1/217             77/6]
[ 1133825/1135924 -2781755/1123584    -58215/205502      -1056/20131      169723/1552]
[    -30305/16636     -104315/1368          505/561           -4/873            165/8]
[       2375/5764        -3055/684        695/13764           -1/339           -29403]
[ -2028915/161732      70027/69084        7073/2275      -2022/79555        -3072/169]


Then, the space of coefficients you are looking for must live inside of the right kernel of the matrix M:

sage: M.right_kernel()
Vector space of degree 5 and dimension 0 over Rational Field
Basis matrix:
[]


If the dimension of the right kernel is 0, then you got your answer. If M.right_kernel() is not of dimension 0, then you should retry again after increasing the number m of rows to make sure this is not an artefact in which case the space of explicit solutions will be given by the basis matrix.

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