My question is about a method described in Dr.Math forum for simplifying equations involving sums of radical functions.
(The following is a transcription of the example given by Dr. Vogler):
--- begin example ---
We want to convert the equation
a + b + c = 0 (eq.1)
a = cuberoot(x) b = fifthroot(y) c = seventhroot(z)
to a polynomial equation without radicals.
a^3 = x b^5 = y c^7 = z
The method is as follows: First multiply eq.1 by every monomial
a^i * b^j * c^k
0 <= i < 3 0 <= j < 5 0 <= k < 7
and replace every occurrence of a^3 by x, of b^5 by y, of c^7 by z. For example, the first few equations would be
a^2 + ab + ac = 0 x + a^2*b + a^2*c = 0 ab + b^2 + bc = 0 a^2*b + ab^2 + abc = 0 ...
Now if you treat x, y, and z as constants and consider only the variables a, b, and c, then you will end up with 3*5*7 polynomial equations in the 3*5*7 monomials
a^i * b^j * c^k.
In other words, if you consider each monomial as a variable, then you can write these in matrix form as
A*v = 0
where A is a square matrix of height and width 3*5*7 which contains the variables x, y, and z but none of a, b, and c, and v is a column vector that lists off the 3*5*7 monomials in a, b, and c. Unless the only solution to your equations is a = b = c = 0, that means that the matrix A is not invertible, and therefore
det(A) = 0.
But the determinant is a polynomial function in the entries of your matrix, which means that the determinant of A is a polynomial in x, y, and z.
--- end example ---
I have tested this method with several examples and it seems to work, at least in the sense that the roots of the radical equation are roots of the resulting polynomial (it seems however that the resulting polynomial may have real roots that are not roots of the original radical equation).
The only method I knew about in order to get rid of radical expressions in equations was to carefully manipulate the equation and then raise both sides of the equation to the same power. This is to say I have a very basic math background.
So finally, my question is: what is the name of this technique and where can I read more about it, or at least what are the keywords I can use to search google.
I have many other questions regarding this technique, but I believe I good book on the subject would answer them. Just for completion, some of the questions are:
Can this technique be used to simplify inequations also? It doesn't look like the resulting polynomial has always the same sign as the original radical expression for the same x interval.
Is there a way to know which roots of the polynomial are roots of the original radical expression without computing and testing them? I can see that the Mathematica "Reduce" function is able to find these polynomials, and then specifically say that my original equation is the first root of the computed polynomial.
I have read something about minimal polynomials, although it looks like they are used for representing algebraic numbers, and not algebraic functions, which I believe is what I have. Nevertheless, the techniques used there look much more complex than this simple trick by Dr. Vogler.