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64 CHAPTER 3. POLYNOMIAL EQUATIONS
Open Problem 1 (Roots of Sparse Polynomials) Find algorithms for counting
the number of real roots which depend polynomially on t alone, or t and
log d. There is recent progress described in [BHPR11]: notably a probabilistic
algorithm when t = 4.
Problem 2 Having solved Problem 1, we may actually wish to know the roots
to a given accuracy, say L bits after the point. Again, many authors have asked
the same question about non-square-free polynomials, and have laboured to
produce better theoretical complexity bounds, since the square-free part of a
polynomial may have larger coefficients than the original polynomial. However,
in practice it is always better to compute the square-free part first.
The best solution to this problem currently is that in [KS11]. TO BE COM-
PLETED
3.2 Linear Equations in Several Variables
We now consider the case of several polynomial equations in several (not necessarily
the same number) of variables.
Notation 16 The variables will be called x 1 , . . . , x n , though in specific examples
we may use x, y or x, y, z etc.
3.2.1 Linear Equations and Matrices
A typical set of 3-by-3 linear equations might look like the following.
2x + 3y − 4z = a;
3x − 2y + 2z = b;
4x − 3y − 2z = c.
⎛
If we denote by M the matrix ⎝ 2 3 −4
⎞
3 −2 2 ⎠, x the (column) vector (x, y, z)
4 −3 −1
and a the (column) vector (a, b, c), then this becomes the single matrix equation
M.x = a, (3.13)
which has, assuming M is invertible, the well-known solution
x = M −1 .a. (3.14)
This poses two questions: how do we store matrices, and how do we compute
inverses?