CC BY
10
УДК 511.42 DOI 10.22405/2226-8383-2016-17-4-180-184
PROBLEM OF NESTERENKO AND METHOD OF BERNIK
N. V. Budarina (Khabarovsk), H. O'Donnell (Dublin)
Dedicated to Yuri Valentinovich Nesterenko and Vasilii Ivanovich Bernik on their 70th birthdays
Abstract
In this article we prove that, if integer polynomial P satisfies |P(w)|p < H-w, then for w > 2n — 2 and sufficiently large H the root 7 belongs to the field of p-adic numbers.
Keywords: integer polynomials, discriminants of polynomials.
Bibliography: 16 titles.
1. Introduction
Throughout this paper, p is a prime number, Qp is the field of p-adic numbers,
P (x) = anxn + ... + a1x + a0
is an integer polynomial with degree deg P(x) = n and height H(P) = max0^.,^ra | aj |. We denote by Vn the set of integer polynomials of degree n. Let Vn(H) = {P eVn : H(P) = H}.
In this paper, a result originally considered by Y. V. Nesterenko is examined. In fl] Y.V. Nesterenko discussed the solvability of the equation P(x) = 0 in the ring of p-adic integers Zp and proved the following result.
Theorem 1. Let x be an integer and P e Vn(H). If
IP(x)lP < е-8"2H-4n, then there exists a p-adic number 7 such that
P(7) = 0, |z — 7|P < 1.
Note that a similar problem was considered in [2] and there was given a criteria for when the closest root of a polynomial to a real point belongs to the field of real numbers. Knowledge of the nature of the roots is very important in the problems of Diophantine approximations for construction of regular systems [3,4]. Numerous applications of this concept arose when obtaining estimates for the Hausdorff measure and Hausdorff dimension of Diophantine sets [5] and proving analogues of the Khintchine theorem [6,7]. Using the regular systems, the exact theorems on approximation of real numbers by real algebraic [6], by algebraic integers [8], of complex numbers by complex algebraic [9] were obtained, and similar problems in the field of p-adic numbers [10] and in R x C x Qp [7] were investigated.
The Theorem 1 can be improved for p-adic leading polynomials. Such a polynomial P eVn satisfies
I On |p » 1. (1)
Theorem 2. Let ш e Zp and P e Vn(H) be a p-adic leading polynomial. Then if
|P<H-w (2)
for w > 2n — 2, and for sufficiently large H > H0(n), it, follows that the root 71 of P belongs to Qp and
|w — 7i|p < 1. (3)
Remark 1. If D(P) = 0 then we have that the root 7^/ P is closest to ш e Zp. The above theorem will be proved using a general method of V.I. Bernik which was developed in [11,12].
2. Preliminary setup and auxilliary Lemmas
Let P £ 'Pre have roots 71,72,... ,Jn in Q*, where Q** is the smallest field containing Qp and all algebraic numbers. Then, from (1) it follows that
|7i|p < 1, i = 1,... ,n; (4)
i.e. the roots are bounded. This follows from Lemma 4 in ( [13], p.85). Define the sets
Tp(lk) = (w £ Zp : |w - 7fc= min |w - 7i|*}, 1 ^ k ^ n.
l^t^'n
Consider the set Tp(^k) for a fixed k and fa ease of notation assume that k = 1. Next, reorder the other roots so that
|7i - 72^ < |7i - 73 |p < ... < |7i - J^p.
Fix e > 0 where e is sufficiently small and suppose that e1 = eN-1 where N = N(n) > 0 is sufficiently large. Let T = [e-1].
For a polynomial P £ Vn(H) define the real numbers pj by
|7i - lj|p = H-pi, 2 < j < n, p2 ^ P3... > Pn.
Define the integers mj, 2 ^ j ^ n, such that
mj — 1 mj
— ^ Pj < > ma ^ ... ^ mn ^ 0.
Further define numbers Si such that
mi+i +... + mn f . .
Si = -—-, (1 ^ l ^ n — 1), Sn = 0.
The first Lemma is a p-adic analogue of the Lemma, which was proved by Bernik in [14] and is a generalisation of Sprindzuk's Lemma ( [13], p.77).
Lemma 1. [15jLetw £ TP(71). Then
|u - 71 |p < mm (|p(u)lplP'(n)|-1 I! |7i - Ik|p)1/j.
^^ k=2
The following Lemma is often referred to as Gelfond's Lemma.
Lemma 2 ( [16], Lemma A.3). Let P1, P2,..., Pk be polynomials of degree n1,... ,nk respectively, and let P = P1P2 ...Pk.Le t n = n1 + n2 + ... + nk. Then
2-nH(P{)H(P2)... H(Pk) < H(P) < 2nH(P1)H(P2)... H(Pk).
In the proof of theorem we will refer to the following statement known as Hensel's Lemma.
Lemma 3 ( [4], p. 134). Let P be a polynomial with coefficients in Zp, let £ = {0 £ Zp and №(£)|p < Then as n ^ to the sequence
4n+1 = 4n P'(Cn) tends to some root ft £ Qp of the polynomial P and
- & < (£)№'(0|p < 1.
3. Proof of Theorem 2
Two cases must be dealt with separately: D(P) = 0 and D(P) = 0.
3.1. Case I: D(P) = 0
First consider a polynomial P £ Vn(H) satisfying D(P) = 0 and (2), and assume that |P'(w)|p ^ IP(w)|p. We will obtain a contradiction. Using (4), we get |P'(w)|p < H-w/2.
It is well known that ID(P)| = j^---, where
A
an-i an-2 ... «0
... 0 \
an-i an—2 ... ai ao 00
0 ... 0 an an-i an_2 ... ai ao
nan (n — 1)an_i (n — 2)a„_2 ... ai . .. 0 ... 0
0 nan (n — 1)a„_i (n — 2)an_2 ... ai 0 ... 0
\ 0 0 ... 0 nan (n — 1)a„_i (n — 2)an_2 ... ai
Hence the determinant,
| A | < |an|((2n — 2)\(nH )2n-2 + n(2n — 2)\(nH )2n-2)
= |an|(2n — 2)!(n + 1)(nH)2n-2 < 2n2n-1(2n — 2)! H2n-2IanI,
using the fact that H < H, i = 0,1,... ,n. Thus, ID(P)| < 2n2n-1(2n — 2)! H2n-2. This implies that
ID(P)|p ^ 2-1n1-2n((2n — 2)!)-1H-2n+2. (5)
Using Lemma 1, |an|p » 1 and (2),
|w — 71|p < mim^ndP(w)|P|P'(71)I-1 ni=2 I71 — 7kIp)1/3
< minKKn(#-w Ian|-^ EIn=j+1 171 — Ik|-1)1/j
< minKKn(#-w M-1^' )1/]
« min1^j^nH j . Define a(P) as the cylinder of points w satisfying
— W + Sj
Iw — 71 Ip « min H .
Let dj = w .S] and denote bv d0 the maximum value of 9j, j = 1,... ,n.
Now the polynomial P' is expanded as a Taylor series and each term is estimated on a(P). Thus
n
P'(u) = P '(71) + Y,((j — 1)!)-1p °°(71)(^ — 71)J-1, =2
|P (j)( 71)(w — 71)i-1|p « H - Si+(n-j) eiH - do(i-1). As d0 ^ dj, this implies that
j—1
|P00(71)(w — 71)^-1|p « H-Sj +(n-)£1+^(-w+Sj) < H-w/2+(n-2)d for 2 <
Thus,
|P'(71)|p < max ||P(%1)(w — 71 y-1 |p} « H-w/2+(n-2)^1
0
a
n
0
a
n
for H > Ho(n).
Expressing the discriminant D(P) in the form
p = KHn-2 n |7t - lj|p = K|pn-V(71)|p n to - 7j|p
and using the facts that |7t |p « 1 and |an |p ^ 1, we obtain
№P)|p (71)|p.
This contradicts (5) for w > 2n-2+2(n-2)e1 and sufficiently large Therefore, |P'(w)|p > |P(w)|p holds forw> 2n - 2 + 2(n - 2)e1; and case I follows immediately from Lemma 3. Hence, there exists a root 71 £ Qp of P such that |w - 71|p ^ |P(w)|p/|P;(w)|^ < 1.
3.2. Case II: D(P) = 0
Consider the polynomial P £ Vn satisfying D(P) = 0. First, P is decomposed into irreducible polynomials Ti(u) £ Z[w], i.e.
k
p m = n fa).
i=1
It will be shown that for some index j, 1 ^ j ^ k,
\Tj(u)l < 2nw/2H-w(Tj). (6)
Assume the opposite, so that
|Tj(w)|p ^ 2nw/2H-w(Tj) for all j, 1 < j < k.
Then, by Lemma 2,
k
^(w)|p ^ n(2nW/2H-W(Tj))S' > 2nw(^=i /2-1)H(P)-w ^ H(P)-w j=1
which contradicts (2). Thus (6) holds.
Hence, applying the same method as in Case I for Tj, D(Tj) = 0, which satisfies (6), it follows
that there exists a p-adic number 71 such th at |w - 71|p < 1 and Tj (71) = 0. This implies P (71) = 0. □
REFERENCES
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Khabarovsk Division of Institute for Applied Mathematics
Dublin Institute of Technology
Получено 28.11.2016 г.
Принято в печать 12.12.2016 г.
