How to eliminate square roots in polynomial?

Question

From the determinant of a matrix $\mathbf M$, I derive a symbolic expression of a polynomial say for example, 1 + a*x^2 + b*x^4 + Sqrt[a^2 - (x + c)^2] - Sqrt[a^2 - (x - c)^2] == 0 . (My actual equation is far more complex than that) The obvious way to eliminate square roots manually is to shift to right hand side the sqrt term and square both side, rearrange and repeat until the all square root terms vanish. This way we preserve equation without any limit approach. Does Mathematica have a function for this?

Answer 1

You can use Eliminate for this. Include auxiliary variables s1 and s2, and then eliminate them:

Eliminate[
    {1 + a x^2 + b x^4 + s1 - s2 == 0, s1^2 == x^2 + c, s2^2 == x^2 - c},
    {s1, s2}
]

-1 + 4 x^2 - 4 a x^2 + 8 a x^4 - 6 a^2 x^4 - 4 b x^4 + 4 a^2 x^6 - 4 a^3 x^6 + 8 b x^6 - 12 a b x^6 - a^4 x^8 + 8 a b x^8 - 12 a^2 b x^8 - 6 b^2 x^8 - 4 a^3 b x^10 + 4 b^2 x^10 - 12 a b^2 x^10 - 6 a^2 b^2 x^12 - 4 b^3 x^12 - 4 a b^3 x^14 - b^4 x^16 == 4 c^2

An alternative usually suggested by @DanielLichtblau is the use of GroebnerBasis:

GroebnerBasis[
    {1 + a x^2 + b x^4 + s1 - s2, s1^2 - x^2 - c, s2^2 - x^2 + c},
    x,
    {s1, s2}
]

{1 + 4 c^2 - 4 x^2 + 4 a x^2 - 8 a x^4 + 6 a^2 x^4 + 4 b x^4 - 4 a^2 x^6 + 4 a^3 x^6 - 8 b x^6 + 12 a b x^6 + a^4 x^8 - 8 a b x^8 + 12 a^2 b x^8 + 6 b^2 x^8 + 4 a^3 b x^10 - 4 b^2 x^10 + 12 a b^2 x^10 + 6 a^2 b^2 x^12 + 4 b^3 x^12 + 4 a b^3 x^14 + b^4 x^16}