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Hölder Continuity of Roots of Complex and p -adic Polynomials

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Abstract

Let f be a polynomial with coefficients in an algebraically closed, valued field. We show a refinement of the principle of continuity of roots, namely, that each root α of f is locally Hölder continuous of order 1/μ as a function of the coefficients of f, where μ is the root multiplicity of α. This is derived as a consequence of a principle that could be called continuity of factors, namely, that if f = gh is a factorisation with (g, h) = 1, then the coefficients of g and h are locally Lipschitz continuous as functions of the coefficients of f. The proofs are elementary and of an algebraic nature.
older continuity of roots of complex and p-adic
polynomials
David Brink
December 2008
Abstract. Let f be a polynomial with coefficients in an algebraically
closed, valued field. We show a refinement of the principle of continuity
of roots, namely that each root α of f is locally older continuous of
order 1 as a function of the coefficients of f , where µ is the root mul-
tiplicity of α. This is derived as a consequence of a principle that could
be called continuity of factors, namely that if f = gh is a factorisation
with (g, h) = 1, then the coefficients of g and h are locally Lipschitz con-
tinuous as functions of the coefficients of f . The proofs are elementary
and of an algebraic nature.
Introduction. Consider a polynomial f over, say, the complex or p-adic num-
bers. Since the coefficients are elementary symmetric functions of the roots, it is
clear that the former depend continuously on the latter. The converse statement
– that the roots depend continuously on the coefficients is a fundamental result
going back at least to Weber’s Lehrbuch [10] in the form stated below.
We show here a refinement of this principle of continuity of roots, namely
that each root α of f is locally older continuous of order 1 as a function of
the coefficients of f, where µ is the root multiplicity of α. This is derived as a
consequence of a principle that could be called continuity of factors, namely that
if f = gh is a factorisation with (g, h) = 1, then the coefficients of g and h are
locally Lipschitz continuous as functions of the coefficients of f. Of course, all
these continuity statements first need a more precise formulation since there is
no natural ordering of the roots of a polynomial.
Continuity of roots is often proved using special properties about C or Q
p
such as Rouch´e’s theorem [7, p. 43], compactness of projective space [3], Newton
polygons [1], or assuming separability of f [5, p. 53]. The arguments used here are
both more elementary and more general. Indeed, they work for any polynomial
over any algebraically closed, valued field (K, | |).
1
By valued field we mean a field equipped with a valuation or a Krull valuation.
A valuation is a map | | : K R
0
satisfying (i) |x| = 0 if and only if x = 0,
(ii) |x · y| = |x| · |y|, and (iii) |x + y| |x| + |y|. It is called non-Archimedean if
(iii
0
) |x+y| max{|x|, |y|}. A Krull valuation is a map | | : K Γ{0} satisfying
(i), (ii) and (iii
0
), where Γ {0} is a totally ordered, divisible, multiplicatively
written Abelian group with zero (cf. [5]).
Results. So let
f = X
n
+ a
n1
X
n1
+ · · · + a
0
=
n
Y
i=1
(X α
i
)
and
f
= X
n
+ a
n1
X
n1
+ · · · + a
0
be two monic polynomials of common degree n over an algebraically closed, valued
field (K, | |). We think of f as fixed and of f
as a variable perturbation of f.
We put
|f
f| := max
0i<n
|a
i
a
i
|
and write f
f if |f
f | 0. Further, we choose an ordering of the roots of
f
by writing
f
=
n
Y
i=1
(X α
i
)
in such a way that |α
i
α
i
| = min
ijn
|α
j
α
i
| for all i. We may then view each
root α
i
as a function of f
. With this notation, continuity of roots can be stated
as α
i
α
i
in the limit f
f for all i. For the sake of completeness, we give a
strikingly simple, but seemingly forgotten proof of this fact due to Coolidge [2].
Theorem 0 (Continuity of roots). Let f =
Q
n
i=1
(X α
i
) be a monic polyno-
mial of degree n over an algebraically closed, valued field (K, | |). Then, given
any > 0, there is a δ > 0 such that any monic polynomial f
of degree n with
|f
f| δ can be written f
=
Q
n
i=1
(X α
i
) such that |α
i
α
i
| for all i.
Proof. From f(α
1
) = 0 follows
Q
n
i=1
(α
1
α
i
) = f
(α
1
) 0 and hence α
1
α
1
in the limit f
f since |α
1
α
1
| is minimal among |α
1
α
i
|. If f = gh is any
factorisation with monic g and h, polynomial division shows that the coefficients
of, say, h can be expressed as polynomials in the the coefficients of f and g. Hence
f
/(X α
1
) f/(X α
1
) in the limit f
f. Now repeat the above argument
to conclude α
2
α
2
, and so forth.
2
If f = gh is any factorisation with monic g and h written as
g =
Y
iI
(X α
i
), h =
Y
jJ
(X α
j
), I J = {1, . . . , n},
then we define the induced factorisation of f
as f
= g
h
where
g
=
Y
iI
(X α
i
), h
=
Y
jJ
(X α
j
).
If follows from Theorem 0 that g
g and h
h in the limit f
f.
For any function γ
= γ
(f
) of f
with γ = γ
(f), we write γ
µ
γ and
say that γ
is older continuous of order 1 in the limit f
f if there are
constants C and > 0 such that
|γ
γ| C · |f
f|
1
for all f
with |f
f| . older continuity of order 1 is also called Lipschitz
continuity.
Theorem 1 (Lipschitz continuity of factors). Let f be a monic polynomial of
degree n over an algebraically closed, valued field (K, | |). Assume there is a
factorisation f = gh with monic polynomials g and h satisfying (g, h) = 1. Then
there are constants C and > 0 such that any monic polynomial f
of degree n
with |f
f| can be written f
= g
h
with monic polynomials g
and h
satisfying |g g
| C · |f
f| and |h h
| C · |f
f|.
Proof. Let f
= g
h
be the induced factorisation as defined above. We show
that g
1
g and h
1
h in the limit f
f.
First assume that g is of the form X
µ
. Write
g
= X
µ
+
µ1
X
i=0
b
i
X
i
, h = X
ν
+
ν1
X
i=0
c
i
X
i
, h
= X
ν
+
ν1
X
i=0
c
i
X
i
where necessarily c
0
6= 0. Obviously, a
i
1
a
i
= 0 for i = 0, . . . , µ 1. As already
noted, h
h by Theorem 0, and thus c
i
c
i
for all i. The factorisation
f
= g
h
gives the relations a
0
= b
0
c
0
, a
1
= b
0
c
1
+ b
1
c
0
, etc. From a
0
1
0 and
c
0
c
0
6= 0 it now follows that b
0
= a
0
/c
0
1
0. Similarly, from a
1
1
0, b
0
1
0,
c
1
c
1
and c
0
c
0
6= 0 it follows that b
1
= (a
1
b
0
c
1
)/c
0
1
0, and so forth for
all b
i
. We have thus shown g
1
X
µ
.
Next assume that g is of the form (X α)
µ
. Translate the polynomials, i.e.
write f(X + α) = g(X + α)h(X + α) and f
(X + α) = g
(X + α)h
(X + α). Then
3
f
(X + α)
1
f(X + α) in the limit f
1
f, hence g
(X + α)
1
g(X + α) = X
µ
by the above, and consequently g
1
g.
Finally, the general case is seen by writing both g and h as products of poly-
nomials of the form (X α
i
)
µ
i
since then (X α
i
)
µ
i
1
(X α
i
)
µ
i
for all i.
Lemma 1 (cf. [8, Theorem 1.1.4]). Any root α of a monic polynomial g =
X
µ
+ a
µ1
X
µ1
+ · · · + a
0
of degree µ satisfies |α| < 2 · max
0i<µ
|a
i
|
1
in the
Archimedean case and |α| max
0i<µ
|a
i
|
1
in the non-Archimedean case.
Proof. Any root α satisfies α
µ
= (a
µ1
α
µ1
+· · · + a
0
). Put = max
0i<µ
|a
i
|
1
and assume first = 1. Then |α|
µ
|α|
µ1
+ · · · + |α| + 1 and therefore |α| < 2
in the Archimedean case, and |α| 1 in the non-Archimedean case. The case of
arbitrary follows from the above by scaling, i.e. replacing g by c
1
· g(c
1
X)
where c is some field element with |c| = .
Theorem 2 (H¨older continuity of roots). Let f =
Q
n
i=1
(X α
i
) be a monic
polynomial of degree n over an algebraically closed, valued field (K, | |). For each
i, let µ
i
be the root multiplicity of α
i
. Then there are constants C and > 0
such that any monic polynomial f
of degree n with |f
f| can be written
f
=
Q
n
i=1
(X α
i
) such that |α
i
α
i
| C · |f
f|
1
i
for all i.
Proof. We show that α
i
µ
i
α
i
for all i. Let i be arbitrary and write f = gh with
g = (X α
i
)
µ
i
. Then (g, h) = 1 and for any f
we have the induced factorisation
f
= g
h
. By Theorem 1, g
1
g and hence g
(X + α
i
)
1
X
µ
i
. It then follows
from Lemma 1 that α
i
α
i
µ
i
0 since α
i
α
i
is a root of g
(X + α
i
), and
consequently α
i
µ
i
α
i
as claimed.
Concluding remarks. A priori, it is more natural to study ordinary continuity
of roots within the more general framework of arbitrary topological fields. This
problem, which turns out to be quite delicate, was treated by Endler [4].
The presumption that K be an algebraically closed field is not essential. If
all polynomials are required to split into linear factors, we may replace K by an
arbitrary integral domain.
It is shown in [8, Remark 1.3.4] that the roots of a complex polynomial of
degree n satisfy a older condition of order 1/n.
In the non-Archimedean case, one has much more refined results. Indeed, the
Newton polygon method gives the exact valuations of the roots of any polynomial
given the valuations of the coefficients of that polynomial. Using this, one can
construct explicit error functions giving best possible bounds on the roots of a
4
perturbed polynomial (see [1]). Similar results were proved by Ershov [6].
Let f
= f
(T, X) C[T, X] and assume that α is a simple root of f =
f
(t
0
, X) for some t
0
C. Then there is an analytic function α
defined in a
neighbourhood U of t
0
such that α
(t
0
) = α and f
(t, α
(t)) = 0 for all t
U. This result on analyticity of roots follows from the complex-analytic implicit
function theorem, but was originally proved by Cauchy in a different manner (see
[9, p. 14] and also [11, p. 39]).
Theorem 1 explains why a small perturbation of the coefficients of a complex
polynomial f typically causes a root α of multiplicity µ to turn into µ simple
roots α
i
arranged more or less symmetrically around α (see also [11, pp. 40–41]).
Simple roots are Lipschitz continuous, but of course they may nonetheless
be sensitive to perturbations of the coefficients if the constant C is very large.
Wilkinson’s polynomial
Q
20
i=1
(X i) (see [11, pp. 42–43] and [12]) is a famous or
even notorious example of this, the discovery of which Wilkinson later described
as “the most traumatic experience in my career as a numerical analyst”!
References
[1] D. Brink, New light on Hensel’s lemma, Expo. Math. 24 (2006), 291–306.
[2] J. L. Coolidge, The continuity of the roots of an algebraic equation, Ann. of
Math. (2) 9 (1908), no. 3, 116–118.
[3] F. Cucker, A. G. Corbalan, An alternate proof of the continuity of roots of
a polynomial, Amer. Math. Monthly 96 (1989), no. 4, 342–345.
[4] O. Endler, On the continuity of the roots of polynomials. (Portuguese).
In: Proc. Fifth Brazilian Math. Colloq., pp. 59–70, Conselho Nacional de
Pesquisas, ao Paulo, 1965.
[5] A. J. Engler, A. Prestel, Valued Fields, Springer, Berlin, 2005.
[6] Yu. L. Ershov, Root continuity theorems in valued fields, Siberian Math. J.
47 (2006), no. 6, 1027–1033.
[7] J. M. Ortega, Numerical Analysis. A Second Course, Academic Press, New
York, 1972.
[8] Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, Clarendon
Press, Oxford, 2002.
5
[9] H. olklein, Groups as Galois Groups, Cambridge University Press, Cam-
bridge, 1996.
[10] H. Weber, Lehrbuch der Algebra. Erster Band, Friedrich Vieweg und Sohn,
Braunschweig, 1895.
[11] J. H. Wilkinson, Rounding Errors in Algebraic Processes, Prentice Hall,
Englewood Cliffs, NJ, 1963.
[12] J. H. Wilkinson, The perfidious polynomial. In: G. H. Golub (ed.), Studies
in Numerical Analysis, pp. 1–28, The Mathematical Association of America,
Washington, DC, 1984.
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