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62 CHAPTER 3. POLYNOMIAL EQUATIONS
Proposition 19 [Sen08, Corollary 3.2] Every irreducible plane curve of degree
at most five is soluble by radicals.
Proposition 20 [Sen08, Corollary 3.3] Every irreducible singular plane curve
of degree at most six is soluble by radicals.
Algorithms to compute these expressions are given in [Har11, SS11].
It is also the case 4 that the offset, i.e. the curve defined as the set of points
a fixed distance d from the original curve, to a curve soluble by radicals is also
soluble by radicals.
3.1.9 How many real roots?
We have seen that it is not obvious how many real roots a polynomial has: can
we answer that question, or more formally the following?
Problem 1 Given a square-free polynomial f, determine how many real roots
f has, and describe each real root sufficiently precisely to distinguish it from
the others. Many authors have asked the same question about non-squarefree
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.
Definition 30 introduced the concept of a signed polynomial remainder sequence,
also called a Sturm–Habicht sequence: f i is proportional by a positive constant
to −rem(f i−2 , f i−1 ). The positive constant is normally chosen to keep the coefficients
integral and as small as possible.
Definition 36 If f is a square-free polynomial, let V f (a) denote the number of
sign changes in the sequence f 0 (a), f 1 (a), . . . , f n (a), where f 0 , . . . , f n is the
Sturm–Habicht sequence of f and f ′ , also known as the Sturm sequence of f.
If f is not square-free, we need more careful definitions [BPR06], and to be clear
whether we are counting with multiplicity or not.
Theorem 11 (Sturm) If a < b are not zeros of f, and f is square-free, then
V f (a) − V f (b) is the number of zeros of f in (a, b).
V f (∞) (which can be regarded as lim a→∞ V f (a)) can be computed as the number
of sign changes in the sequence of leading coefficients of the Sturm sequence.
4 Unpublished. Prof. Sendra has supplied this proof for plane curves.
Let (R 1 , R 2 ) be a square parametrization of the curve and K 0 = C(t) ⊂ K 1 ⊂ . . . ⊂ K s be
a (radical) field tower such that R 1 , R 2 ∈ K s. Considering the formal derivation with respect
to t, one can deduce (for instance by induction on s) that if R ∈ K s then its derivative R ′ is
also in K s.
Now consider a = (R ′ 1 )2 + (R ′ 2 )2 ∈ K s and K s+1 = K s( √ a), then (O 1 , O 2 ) = (R 1 , R 2 ) ±
d/ √ a(−(R 2 ) ′ , (R 1 ) ′ ) ∈ (K s+1 ) 2 . So (O 1 , O 2 ) is radical with the tower K 0 ⊂ . . . ⊂ K s ⊂
K s+1 .