The Abhyankar - Moh theorem

The Abhyankar - Moh theorem

Andrzej Nowicki

N. Copernicus University, Faculty of Mathematics and Informatics, 87–100 Toru´n, Poland

(e-mail: anow@mat.uni.torun.pl) November 2, 1997

In this note we present a proof of the following theorem.

Theorem 0.1. Let k[t] be the ring of polynomials in one variable t over a field k of charac- teristic zero, and let f, g ∈ k[t] r k. If k[f, g] = k[t], then deg f | deg g or deg g | deg f .

The above theorem appeared in the fifties with an incorrect proof in Segre’s paper [12]

which discusses a subject of the Jacobian conjecture. The first time it was proved in 1975 by Abhyankar and Moh [3] (see also [1]). The history of this theorem and its applications are presented in the papers [3] and [4]. There appeared also other proofs in the years 1977 – 1982. They were publicated by Miyanishi [7] (see also [8]), Ganong [5] and Rudolph [11].

Abhyankar and Moh proved this theorem basing on the theory of approximate roots of polynomials which was developed by them. In 1986 Richman [10] presented a new interesting proof which does not use approximate roots. There is however a certain gap in his proof, noticed and corrected, in 1991, by Kang [6] (see also [9]).

The purpose of this article is a presentation of the full proof of Theorem 0.1 according to Richman.

We assumed in Theorem 0.1 that k is a field of characteristic zero. With a certain additional assumption which is ”char(k) - gcd(deg f, deg g)”, this theorem is also true for fields of positive characteristics (see for example [3]). In such version Richman proves it in [10]. But we do not take up this case here.

We stress that it is easy to prove Theorem 0.1 when the degree of f or g is a prime number (see Corollary 1.7).

The author would like to thank Ilona Nowosad and Maciej Smoczy´nski for their help in the preparation of this note.

1 Preliminary facts

Throughout this article k is a field of characteristic zero, k[t] is the polynomial ring in one variable t over k and k(t) is the field of rational functions in the variable t over k. If K ⊆ L is a finite field extension, then we denote by (L : K) the degree of this extension.

Proposition 1.1. If R is a ring such that k ( R ⊆ k[t], then the ring k[t] is integral over R.

Proof. Let f = a_ntⁿ+ · · · + a₁t + a₀ where a_n. . . , a₀ ∈ k, a_n 6= 0, be a polynomial belonging to R r k. Then we have the equality tⁿ+ a⁻¹_n a_n−1tⁿ⁻¹+ · · · + a⁻¹_n (a₀− f ) = 0, which implies that t is integral over R (even over k[f ]). 

In a similar way we prove:

Proposition 1.2. If g ∈ k[t] r k, then k(g) ⊆ k(t) is a finite field extension and (k(t) : k(g)) = deg g. 

Assume now that f and g are polynomials belonging to k[t] r k, and consider the rings k ( k[g] ⊆ k[f, g] ⊆ k[t] and the fields k ( k(g) ⊆ k(f, g) ⊆ k(t). Proposition 1.1 implies that k[f, g] is integral over k[g]. In particular, the polynomial f is integral over k[g]. Therefore there exists a polynomial W ∈ k[g][X] such that W (f ) = 0.

Proposition 1.3. Let f, g ∈ k[t] r k and let W ∈ k[g][X] be the monic polynomial of the minimal degree such that W (f ) = 0. Then deg W = (k(f, g) : k(g)).

Proof. Let N = (k(f, g) : k(g)) = (k(g)(f ) : k(g)) and let B ∈ k(g)[X] be a minimal polynomial for f over k(g). Since W (f ) = 0 and W ∈ k[g][X] ⊂ k(g)[X], we obtain W = AB for some A ∈ k(g)[X]. Let a, b be elements from k[g] such that aA, bB ∈ k[g][X]. Then in the ring k[g][X] we have the equality: abW = aA · bB.

The ring k[g] is a unique factorization domain (because k[g] is a ring isomorphic to k[t]).

Therefore there exist elements a1, b1 ∈ k[g] and primitive polynomials A₁, B1 ∈ k[g][X] such that aA = a1A1, bB = b1B1. Then we have the equality:

abW = a₁b₁(A₁B₁).

The polynomial W is primitive (because it is monic) and the polynomial A1B1is also primitive (by Gauss Lemma). Hence W = c(A₁B₁) where c is an invertible element of k[g]. Thus there exists an invertible element d ∈ k[g] such that dB₁is a monic polynomial belonging to k[g][X].

Let H = dB1. Then H ∈ k[g][X] is a monic polynomial, H(f ) = 0, deg H = deg B = N and so, the minimality of W implies that deg W = N = (k(f, g) : k(g)). 

Corollary 1.4. Let f, g ∈ k[t] r k. Then

k[f, g] = k[g]f^{N −1}+ k[g]f^{N −2}+ · · · + k[g]f + k[g], where N = (k(f, g) : k(g)).

Proof. Let E = k[g]f^{N −1}+· · ·+k[g]f +k[g]. We know, by Proposition 1.3, that f^N ∈ E.

It is easy to show (using a simple induction) that fⁿ∈ E for each n > N.  Corollary 1.5. Let f, g ∈ k[t]. If k(f, g) = k(g), then k[f, g] = k[g].

Proof. It follows from Corollary 1.4 for N = 1.  Corollary 1.6. If g ∈ k[t], then k(g) ∩ k[t] = k[g].

Proof. Let f ∈ k(g) ∩ k[t]. Then k(f, g) = k(g) and so, by Corollary 1.5, k[f, g] = k[g].

Thus f ∈ k[g]. 

The next corollary is a special case of Theorem 0.1.

Corollary 1.7. Let f, g ∈ k[t] rk. Assume that one of the numbers deg f and deg g is prime.

If k[f, g] = k[t], then deg f | deg g or deg g | deg f .

Proof. Let deg g = p, where p is prime. Then (by the assumption and Proposition 1.2) (k(f, g) : k(g)) = 1 · (k(f, g) : k(g))

= (k(t) : k(f, g))(k(f, g) : k(g))

= (k(t) : k(g))

= deg g = p,

and hence k[f, g] = k[g]f^p−1+ · · · + k[g]f + k[g] (Corollary 1.4). Since t ∈ k[f, g], there exist nonzero elements a₁, . . . , a_s∈ k[g] such that

t = a1fⁱ¹ + · · · + asf^is, where s > 1 and p > i1 > · · · i_s> 0.

Assume that deg g - deg f . Then the degrees of the polynomials a1fⁱ¹, . . . , a_sf^is are pairwise incongruent modulo p (because the degrees of the polynomials a1, . . . , asare divisible by p and p - deg f ). Therefore there exists r ∈ {1, . . . , s} such that ir > 0 and 1 = deg t = deg(a_rf^ir). This implies that deg f = 1 and we have: deg f | deg g. 

2 Notations and definitions

For the rest of the article we assume that f and g are polynomials belonging to k[t] r k.

We denote by N the number (k(f, g) : k(g)). If N = 1, then (see Corollary 1.5) f ∈ k[g] and then deg g | deg f . Therefore we assume that N > 1. The equalities

deg g = (k(t) : k(g)) = (k(t) : k(f, g))(k(f, g) : k(g)) imply that N 6 deg g.

If n is a positive integer then we denote:

An = fⁿ+ k[g]fⁿ⁻¹+ · · · + k[g]f + k[g], A_n = fⁿ+ k(g)fⁿ⁻¹+ · · · + k(g)f + k(g), L_n = k[g]fⁿ+ k[g]fⁿ⁻¹+ · · · + k[g]f + k[g], Ln = k(g)fⁿ+ k(g)fⁿ⁻¹+ · · · + k(g)f + k(g).

Moreover, we denote: L₀= k[g] and L₀= k(g). Observe that A_n= fⁿ+ L_n−1, A_n= fⁿ+ L_n−1.

If n > N then An = k[f, g] and An = k(f, g) (see Corollary 1.4). If n > N − 1 then L_n= k[f, g] and L_n= k(f, g).

Proposition 2.1. If n is a positive integer, then An∩ k[f, g] = A_n and Ln∩ k[f, g] = L_n.

Proof. We will show that A_n∩ k[f, g] = A_n. If n > N then the equality is obvious because in this case A_n = k(f, g). Assume now that n < N and let u ∈ A_n ∩ k[f, g].

Then u = fⁿ+ an−1fⁿ⁻¹+ · · · + a1f + a0, for some an−1, . . . , a0 ∈ k(g). On the other hand u = b_{N −1}f^{N −1} + · · · + b₁f + b₀, where b_{N −1}, . . . , b₀ ∈ k[g], because u ∈ k[f, g] = k[g]f^{N −1}+ · · · + k[g]f + k[g]. The polynomials f^{N −1}, . . . , f¹, f⁰ form a basis of the space k(g)(f ) over k(g). The representation of the element u is then unique. This implies that the elements a_n−1, . . . , a₀ belong to k[g], thus u ∈ A_n. Therefore we have the inclusion A_n∩ k[f, g] ⊆ A_n. The opposite inclusion is obvious. In a similar way we prove that Ln∩ k[f, g] = L_n. 

Definition 2.2. A sequence (h₁, . . . , h_{N −1}) of elements of the field k(f, g) is called an α- system, if:

(a) h_n∈ A_n for each n = 1, . . . , N − 1,

(b) the numbers 0, deg h1, . . . , deg h_{N −1} are pairwise incongruent modulo deg g.

Definition 2.3. A sequence (h₁, . . . , h_{N −1}) of elements of the ring k[f, g] is called a β-system, if:

(a) hn∈ A_n for each n = 1, . . . , N − 1,

(b) the numbers 0, deg h1, . . . , deg hN −1 are pairwise incongruent modulo deg g.

3 Properties of α and β-systems

Every α-system (h1, . . . , hN −1), such that h1, . . . , hN −1∈ k[f, g], is (by Proposition 2.1) a β-system. It is easy to prove the following two propositions.

Proposition 3.1. Let (h1, . . . , h_{N −1}) be an α-system and let n ∈ {1, . . . , N − 1}. Then:

(1) An= hn+ Ln−1,

(2) L_n= k(g)h_n+ · · · + k(g)h₁+ k(g). 

Proposition 3.2. Let (h₁, . . . , h_{N −1}) be a β-system. Then L_n= k[g]h_n+ · · · + k[g]h₁+ k[g]

for all n ∈ {1, . . . , N − 1}.  Now we prove:

Proposition 3.3. Let (h1, . . . , hN −1) be a β-system. Let h0 = 1, n < N and let w ∈ Ln. Then there exist uniquely determined integers s and r such that deg w = deg g^sh_r, s > 0 and r ∈ {0, 1, . . . , n}.

Proof. We know from the previous proposition that Ln= k[g]hn+ · · · + k[g]h1+ k[g]h0. Thus there exist polynomials un, . . . , u0 ∈ k[g] such that w = u_nhn+ · · · + u0h0. From the definition of β-system it follows that the degrees of the polynomials u_nh_n, . . . , u₀h₀ are pairwise distinct. Therefore deg w = deg u_rh_r, for a certain r ∈ {0, 1, . . . , n}. The number deg ur is divisible by deg g (because ur ∈ k[g]). Thus there exists a non-negative integer s satisfying the equality deg u_r = s deg g. Then we have: deg w = deg u_rh_r= deg g^sh_r.

Suppose now that deg g^s1h_r1 = deg g^s2h_r2 for some integers r₁, s₁, r₂, s₂such that s₁, s₂ >

0 and r1, r2∈ {0, . . . , n}. Then

deg h_r1 − deg h_r2 = (s₁− s₂) deg g,

and hence deg hr1 ≡ deg h_r2 (mod deg g). Therefore r1 = r2 and s1 = s2.  The following theorem play an essential role in the proof of Theorem 0.1.

Theorem 3.4. There exists a β-system.

A proof of this theorem we will present in Section 10.

4 Proof of Theorem 0.1

Proof. Denote: d_f = deg f , dg = deg g, d = gcd(d_f, dg). Let d_f = ad, dg = bd where a and b are relatively prime natural numbers. There exist integers m and n such that 1 = ma + nb and 0 6 m < b. Then d = mdf + ndg, and therefore

d ≡ deg f^m (mod d_g). (4.1)

Let us recall that we denote by N the number (k(f, g) : k(g)). In our case: N = d_g (because d_g = (k(t) : k(g)) = (k(t) : k(f, g))N = 1 · N = N ).

Let (h₀ = 1, h₁, . . . , h_{N −1}) be a β-system. Since the polynomial t^d belongs to k[f, g] = LN −1 (see Corollary 1.4), then, by Proposition 3.3, there exist integers r and s such that

d = deg t^d= deg g^sh_r, s > 0 i r ∈ {0, 1, . . . , N − 1}. (4.2) The polynomial f^m belongs to L_m. Applying Proposition 3.3 once more (this time for the polynomial f^m) we observe that deg f^m = deg g^s1hr1, where 0 6 r16 m and s1 > 0.

From (4.1) and (4.2) it follows that deg hr1 ≡ deg h_r (mod dg) and consequently r1= r.

Therefore we have the inequality r < b which implies that hr∈ L_b−1= k[g]f^b−1+ · · · + k[g]f⁰. Observe now that the degrees of the polynomials f⁰, f¹, . . . , f^b−1are pairwise incongruent modulo d_g. In fact, suppose that deg fⁱ ≡ deg f^j (mod d_g) for some i, j ∈ {0, 1, . . . , b − 1}.

Then (i − j)d_f = ud_g, where u ∈ Z, and we have the equality (i − j)a = ub which implies that b | i − j (because the numbers a and b are relatively prime), and so i = j.

Repeating the proof of Proposition 3.3 we see that deg hr= deg g^pfⁱ where p, i are some non-negative integers. Thus we have

d = deg g^sh_r= sd_g+ deg h_r= sd_g+ deg g^pfⁱ= (s + p)d_g+ id_f, where s + p > 0, i > 0 and s + p + i > 0. Consequently:

d = (s + p)d_g+ id_f > min(df, d_g) > gcd(df, d_g) = d, i.e. gcd(d_f, d_g) = min(d_f, d_g). Therefore d_f | d_g lub d_g | d_f. 

In the above proof the assumption ”k[t] = k[f, g]” we use only for a statement, that there exists a nonzero polynomial w belonging to k[f, g] such that deg w = d, where d = gcd(deg f, deg g). In our case w = t^d. Therefore we have proved:

Theorem 4.1 ([10]). Let f, g ∈ k[t] r k and let d = gcd(deg f, deg g). If k[f, g] contains a nonzero polynomial of the degree d then deg f | deg g or deg g | deg f . 

In the proof we have applied Theorem 3.4 saying that there exists at least one β-system.

This fact we have though not proved yet. The remaining part of the article leads to the proof of this fact.

5 The existence of α-systems

In this section we will prove that there exists at least one α-system.

Let S be a subset of k[t] r {0} such that each two different elements of S have different degrees. Denote by deg S the set {deg s; s ∈ S}. Recall that deg 0 = −∞.

If h ∈ k[t], then we denote by R(h, S) a polynomial belonging to k[t], which we define in the following inductive way.

Definition 5.1.

(1) R(0, S) = 0.

(2) Let h 6= 0, deg h = m > 0 and assume that R(h⁰, S) is already determined for all h⁰∈ k[t] such that deg h⁰ < deg h. Then:

(a) If deg h 6∈ deg S, then we put: R(h, S) = h.

(b) If deg h ∈ deg S, then there exists a unique element s ∈ S such that deg h = deg s and moreover, there exists a unique nonzero element a ∈ k such that deg(h − as) < deg h. In this case we put: R(h, S) = R(h − as, S).

Example 5.2. Let S = {x²+ 1, 2x + 1}. Then deg S = {2, 1} and we have:

(1) R(x³, S) = x³, R(x⁵+ 3x, S) = x⁵ + 3x (because the degrees of the polynomials x³ and x⁵+ 3x do not belong to deg S),

(2) R(x²+ x, S) = R((x²+ x) − (x²+ 1), S) = R(x − 1, S) = R((x − 1) − 1/2(2x + 1), S) = R(−3/2, S) = −3/2. 

Therefore we have the function R( , S) : k[t] −→ k[t]. It is easy to prove:

Proposition 5.3. Let h ∈ k[t]. Then:

(1) deg R(h, S) 6∈ deg S;

(2) if deg h 6∈ deg S, then R(h, S) = h;

(3) R(h, S) = 0 ⇐⇒ h = a1s1+ · · · + apsp where a1, . . . , ap ∈ k i s₁, . . . , sp∈ S.  Now we will prove:

Proposition 5.4 ([10]). There exist polynomials h₁, . . . , h_{N −1}∈ k[t] such that:

(1) hn∈ L_nr Ln−1, for n = 1, . . . , N − 1;

(2) the numbers 0, deg h1, . . . , deg hN −1 are pairwise incongruent modulo deg g.

Proof. We construct the polynomials h1, . . . , hN −1 using an induction.

Let S = {gⁿ; n = 0, 1, . . . }. Let h1 = R(f, S). Then h1 ∈ L₁r L0 and the numbers 0 and deg h₁ are incongruent modulo deg g. Assume that the polynomials h₁, . . . , h_n are already constructed. If n + 1 = N , then the construction is finished. Assume therefore that n + 1 < N and let h0 = 1. Consider the set

T = {hrgⁱ; r = 0, . . . , n, i > 0}.

It is clear that T ⊂ k[t] r {0} and that the elements of T have pairwise different degrees. Let y0= R(fⁿ⁺¹, T ).

Then y₀ ∈ fⁿ⁺¹ + L_n (in particular, y₀ 6= 0) and the elements of the set T ∪ {y₀} (see Proposition 5.3) have pairwise different degrees. Thus we can consider the polynomial

y₁ = R(gfⁿ⁺¹, T ∪ {y₀}).

This polynomial belongs to the set (g + a₀)fⁿ⁺¹+ L_n for some a₀ ∈ k. In particular y₁ 6= 0.

Moreover, the elements of T ∪ {y₀, y₁} have pairwise different degrees.

Continuing this proceeding we can construct an infinite sequence y0, y1, . . . of nonzero polynomials belonging to k[t] such that

yi = R(gⁱfⁿ⁺¹, T ∪ {y0, . . . , yi−1}) for i = 0, 1, . . . .

Now it is easy to show that the degrees of the polynomials from T ∪ {y0, . . . , yi} are pairwise different. In particular, the degree of each of the polynomials y₀, y₁, . . . is not divisible by deg g. It is also easy to verify that yi ∈ (gⁱ+ ai−1gⁱ⁻¹+ · · · + a0)fⁿ⁺¹+ Ln, for some a0, . . . , ai−1∈ k. This implies that the polynomials y₀, y1, . . . belong to Ln+1r Ln.

Now we will prove that among the polynomials y₀, y₁, . . . there exists at least one whose degree is incongruent modulo deg g to any of the numbers deg h0, deg h1, . . . , deg hn .

Assume that it is not true. Then for every non-negative integer j there exists a number m_j ∈ {0, . . . , n} such that

deg yj ≡ deg h_mj (mod deg g).

Since the set {0, . . . , n} is finite, there exists an infinite subset U of non-negative integers and there exists s ∈ {0, . . . , n} such that deg y_u≡ deg h_s (mod deg g) for all u ∈ U . Then

deg yu+ pudeg g = deg hs, where pu ∈ Z. (5.1) Observe that pu > 0. In fact, suppose that pu 6 0. Then deg yu = deg(g^−puhs), g^−puhs∈ T and we have a contradiction because the degrees of the elements of T ∪ {y0, . . . , yu} are pairwise different. Thus every integer of the form p_u is positive. Now the equality (5.1) implies that deg yu < deg hs for all u ∈ U . But it is a contradiction, because the set U is infinite and the degrees of the polynomials y0, y1, . . . are pairwise different.

The obtained contradiction proves that there exists a non-negative integer j such that the numbers deg yj, 0, deg h1, . . . , deg hn are pairwise incongruent modulo deg g. We define now the polynomial hn+1 putting hn+1= yj. 

Let h₁, . . . , h_{N −1} be such polynomials as in the above proposition. The fact that h_n ∈ L_nr Ln−1 (for n = 1, . . . , N − 1) implies that h_n has a nonzero coefficient at fⁿ, belonging to k[g]. Dividing hn by this nonzero coefficient we obtain a rational function h⁰_n belonging to A_n. The numbers 0, deg h⁰₁, . . . , deg h⁰_{N −1} are obviously pairwise incongruent modulo deg g.

Thus we have:

Proposition 5.5. There exists at least one α-system. 

6 Definitions of the sequence c(1), ..., c(p)

Let us recall (see Section 2) that if n = 0, 1, . . . , N − 1, then we denote by Ln the set k(g)fⁿ+ · · · + k(g)f¹+ k(g)f⁰. Let us introduce also the following two notations.

Un = {deg ϕ; ϕ ∈ L_n},

G_n = the ideal in Z generated by Un.

We have then the sequence of ideals G₀ ⊆ G₁ ⊆ · · · ⊆ G_{N −1}. Observe that G₀ is a principal ideal generated by deg g. Moreover we have

Proposition 6.1. G₀6= G₁.

Proof. We know by Proposition 5.4 that there exists h₁ ∈ L₁rL0such that deg g - deg h1. This means that deg h₁ ∈ G₁r G0. 

Now we define a sequence c(1), c(2), . . . , c(p) of certain natural numbers.

Definition 6.2.

(1) c(1) = 1.

(2) Assume that the numbers c(1), . . . , c(m) are already determined and let S be the set of all natural numbers n ∈ {c(m) + 1, c(m) + 2, . . . , N − 1} such that G_c(m)( Gn. If S = ∅, then we end the definition and the number m we denote by p. If S 6= ∅, then c(m + 1) is the smallest element of S.

Moreover, we fix: c(p + 1) = N .

The sequence c(1), . . . , c(p) has the following properties:

(i) c(1) = 1,

(ii) c(1) < c(2) < · · · < c(p) 6 N − 1, (iii) G₀ ( Gc(1)( · · · ( Gc(p),

(iv) G_c(n)= G_c(n)+1 = · · · = G_c(n+1)−1, for n = 1, . . . , p.

We will prove now that the sequence c(1), . . . , c(p) can be defined also in a different way;

using an α-system. For that purpose first prove the following proposition.

Proposition 6.3. Let (h₁, . . . , h_{N −1}) be an α-system and let n ∈ {1, . . . , N − 1}. Denote h0= g. Then:

(1) Un=Sn

j=0{deg h_j+ Z deg g}, (2) G_n= (deg h₀, deg h₁, . . . , deg h_n).

Proof. (1) Let 0 6 j 6 n, s ∈ Z. Then deg hj + s deg g = deg(g^shj) ∈ Un, and thus

∪ⁿ_j=0{deg h_j+ Z deg g} ⊆ Un. The reverse inclusion follows from the fact that all the numbers deg g, deg h₁, . . . , deg h_{N −1} are pairwise incongruent modulo deg g.

(2) It follows from (1) and Proposition 3.1. 

Let (h1, . . . , hN −1) be an α-system. Put h0 = g and denote:

d₀ = deg h₀= deg g d1 = deg h1

...

dN −1 = deg hN −1.

The numbers d0, d1, . . . , dN −1 are integers (they can be negative) and none of them are congruent modulo d0 = deg g. Now the sequence c(1), . . . , c(p) can be defined as follows.

Definition 6.4.

(1) c(1) = 1.

(2) Assume that the numbers c(1), . . . , c(m) are already defined. If the numbers d₀, d₁, . . . , dN −1 belong to the ideal (d0, d1, . . . , d_c(m)), then c(m) is the last element of the sequence and in this case the number m we denote by p. In the opposite case c(m + 1) = i, where i is the smallest of the numbers c(m) + 1, c(m) + 2, . . . , N − 1 such that d_i 6∈ (d₀, d₁, . . . , d_c(m)).

We see (by Proposition 6.3) that the above two definitions describe the same sequence c(1), . . . , c(p). By Definition 6.2 it follows that this sequence does not depend on a selection of an α-system.

Note also an obvious proposition.

Proposition 6.5. Let n ∈ {1, . . . , N − 1}. The following conditions are equivalent:

(1) n appears in the sequence c(1), . . . , c(p);

(2) there exists ϕ ∈ An such that deg ϕ 6∈ Gn; (3) Gn−16= G_n. 

7 The numbers e

, ..., e

Assume that (h₁, . . . , h_{N −1}) is a fixed α-system. Put h₀ = g and let d_i = deg h_i for i = 0, 1, . . . , N − 1. In the previous section we defined the number p and the sequence c(1), . . . , c(p). Additionally we assume that c(p + 1) = N .

Consider now the ideals D₀, D₁, . . . , D_p, of the ring Z, defined as follows:

Definition 7.1.

D_j =

( (d0), for j = 0,

(d0, d1, . . . , d_c(j)), for j ∈ {1, . . . , p}.

Let s ∈ {1, . . . , p}. We know (see Definition 6.4) that d_c(s) 6∈ D_s−1. Since the ideal Ds−1

is principal (because every ideal of Z is principal), there exists a natural number n such that nd_c(s) ∈ D_s−1.

Definition 7.2. We denote by es the smallest natural number n such that nd_c(s)∈ D_s−1. Thus, we have natural numbers e1, . . . , ep. They are greater than 1. We will prove:

Proposition 7.3 ([10]). If s ∈ {1, . . . , p}, then e_sc(s) = c(s + 1).

Before the proof of this proposition we will introduce a new set B_s and we will prove several lemmas.

Let s be an element of the set {1, . . . , p}. Denote:

Bs= {hnh^j_c(s) ; 0 6 n < c(s), 0 6 j < es}.

In particular we have:

B₁ = {g = gh⁰₁, gh¹₁, . . . , gh^e₁¹⁻¹}.

Lemma 7.4.

(1) The degrees of the elements of Bs are pairwise incongruent modulo d0 = deg g.

(2) |Bs| = e_sc(s) (where |Bs| denotes the cardinality of the set B_s).

(3) The set Bs is linearly independent over k(g).

(4) esc(s) 6 N .

(5) The set B_s is a basis of the linear space L_esc(s)−1 over k(g).

Proof. (1). Suppose that

deg(hnh^a_c(s)) ≡ deg(hmh^b_c(s)) (mod d0),

where n, m ∈ {0, 1, . . . , c(s) − 1} and 0 6 a, b < es. If a = b, then deg hn ≡ deg h_m modd0

and hence, by the definition of an α-system, n = m. Therefore a 6= b.

Let a > b. Then (a − b)d_c(s) ∈ D_s−1 and we have a contradiction with the property of the minimality of the number es. The same argument we use in the case b > a.

(2). It follows from (1) that the elements of the form h_nh^j_c(s) (where 0 6 n < c(s) and 0 6 j < es) are pairwise different. Of course there are c(s)es such elements.

(3). Let α₁a₁ + · · · + α_na_n = 0, where α₁, . . . , α_n ∈ k(g) and a₁, . . . , a_n are pairwise different elements of B_s.

Suppose that α1 6= 0. Then a₁ = −α⁻¹₁ α2a2− · · · − α⁻¹₁ αnan. Since the degrees of the elements of the form α⁻¹₁ α_i are divisible by d₀, the number deg a₁ is, by (1), equal to one of the numbers deg(α⁻¹₁ α₂a₂), . . . , deg(α⁻¹₁ α_na_n). But it is a contradiction with (1).

(4). It follows from (2), (3) and from the fact that Bs ⊆ k(f, g) and N = (k(f, g) : k(g)).

(5). If 0 6 n < N , then the polynomials f⁰, f¹, . . . , fⁿare linearly independent over k(g).

Therefore the dimension over k(g), of every space of the form Ln, is equal to n + 1. Since esc(s) − 1 < N (see (4)), we have:

dim_k(g)L_esc(s)−1 = esc(s).

We know from (2) and (3) that the dimension of the subspace generated by Bs is also equal to esc(s). Therefore, it suffices to prove that Bs⊆ L_esc(s)−1.

Let u = hnh^j_c(s), where 0 6 n < c(s) and 0 6 j < es. Then we have:

u ∈ A_n(A_c(s))^j ⊆ A_n+jc(s)⊆ L_n+jc(s)⊆ L_esc(s)−1, and so B_s⊆ L_esc(s)−1. 

Lemma 7.5. esc(s) 6 c(s + 1).

Proof. If s = p, then the inequality follows from Lemma 7.4(4) (because c(p + 1) = N ). Assume now that s < p and let m be a natural number smaller than esc(s). Then hm ∈ L_esc(s)−1 and so, by Lemma 7.4(5), hm = α1a1+ · · · + αnan, where α1, . . . , αn ∈ k(g) and a₁, . . . , a_n are pairwise different elements of B_s. We know, by Lemma 7.4(1), that the elements a1, . . . , as have pairwise incongruent degrees modulo d0. This implies that dm= deg hm = deg(αjaj), for some j ∈ {1, . . . , n}. It is obvious that deg(αjaj) ∈ Ds. Hence d_m ∈ D_s. So, if m < e_sc(s), then d_m ∈ D_s. By the definition of the sequence c(1), . . . , c(p) we know that d_c(s+1) 6∈ D_s. Therefore c(s + 1) > esc(s). 

Lemma 7.6. For every w ∈ D_s there exists b ∈ B_s such that w ≡ deg b (mod d₀)).

Proof. (Induction with respect to s). Let s = 1. Recall that c(1) = 1, B₁ = {gh⁰₁ = g, gh¹₁, . . . , gh^e₁¹⁻¹} and D₁ = (d₀, d₁). Let w ∈ D₁. Then there exist integers a and m such that w = ad0 + md1. Since e1d1 ∈ (d₀), we may assume that 0 6 m < e1. Then w = a deg g + deg h^m₁ = (a − 1) deg g + deg(gh^m₁ ) which implies that w ≡ b (mod d₀), where b = gh^m₁ ∈ B₁.

Assume now that the lemma is true for some s ∈ {1, . . . , p − 1} and let w ∈ Ds+1. Since D_s+1 = D_s+ (d_c(s+1)), we have w = w⁰+ md_c(s+1), where w⁰ ∈ D_s and m ∈ Z. We know that e_s+1d_c(s+1) ∈ D_s. Therefore we may assume that 0 6 m < es+1. Moreover, by the induction, w⁰ ≡ deg b⁰ (mod d0) for some b⁰ ∈ B_s. Using Lemmas 7.4, 7.5 and Proposition 3.1 we obtain:

b⁰ ∈ B_s⊆ L_esc(s)−1 ⊆ L_c(s+1)−1= k(g)h_c(s+1)−1+ · · · + k(g)h₀.

Therefore, deg b⁰ ≡ deg h_r (mod d0), for some r < c(s + 1) (because h0, . . . , h_c(s+1)−1 have pairwise incongruent degrees). Now we have:

w = w⁰+ md_c(s+1) ≡ deg b⁰+ deg h^m_c(s+1) ≡ deg h_r+ deg h^m_c(s+1) = deg b, where b = h_rh^m_c(s+1) ∈ B_s+1. 

Now we are ready to prove the announced proposition.

Proof of Proposition 7.3. Let n = e_sc(s). Then (Lemma 7.4) n 6 N . Assume that n = N and suppose that s < p. Then (by Lemma 7.5) we obtain the following contradiction:

N = esc(s) 6 c(s + 1) 6 c(p) 6 N − 1. Thus, if n = N , then s = p and epc(p) = N = c(p + 1).

Assume now that n < N . We will show that dn 6∈ D_s. Suppose that it is not true.

Let d_n ∈ D_s. Then there exists an element b ∈ B_s (see Lemma 7.6) such that d_n ≡ deg b (mod d0). But, by Lemma 7.4 and Proposition 3.1,

B_s⊂ L_esc(s)−1 = L_n−1= k(g)h_n−1+ · · · + k(g)h₀,

thus deg b ≡ deg hj for some j < n, and hence deg hn = dn ≡ deg h_j. Now we have a contradiction with the fact that the polynomials h0, . . . , hnhave pairwise incongruent degrees.

Therefore d_n6∈ D_s. Thus n > c(s + 1) (see Definition 6.4), esc(s) > c(s + 1) and so, by Lemma 7.5, esc(s) = c(s + 1). 

Corollary 7.7. If s ∈ {1, . . . , p}, then e1e2. . . es = c(s + 1).

Proof. e1 = e1c(1) = c(2), e1e2 = c(2)e2= c(3), etc. 

8 The rational functions of the form w(H, s)

From Section 5 we aim at the proof that there exists a β-system. This fact is essential in the proof of Abhyankar–Moh theorem which is presented in Section 4. We already know (see Section 5) that there exists an α-system.

In the present section with every α-system H we associate certain, uniquely determined, rational functions w(H, 1), . . . , w(H, p), belonging to the spaces Lc(1)−1, . . . , L_c(p)−1, re- spectively. We will prove that the degrees of these functions are lower than the numbers d_c(1), . . . , d_c(s), respectively. This will allow us to prove (in the next section) that there exists

such an α-system for which all the above functions are equal zero. This fact will be very useful in the proof of the theorem on the existence of a β-system.

Let H = (h1, . . . , hN −1) be an α-system and let s be a fixed natural number belonging to the set {1, . . . , p}. We assume additionally that h₀= g and h_N = 0. Before we introduce the rational function mentioned above, we will prove the following two propositions.

Proposition 8.1. For each w ∈ L_c(s+1)−1 there exist uniquely determined elements w1, w2, . . . , w_e, belonging to L_c(s)−1, such that

w = w₁h^e−1+ w₂h^e−2+ · · · + w_e−1h¹+ w_eh⁰, (8.1) where h = h_c(s) and e = es.

Proof. As in Section 7, let B = B_s= {h_nh^j; 0 6 n < c(s), 0 6 j < e}. We know that B is a basis of the space L_ec(s)−1 = L_c(s+1)−1over k(g) (see Lemma 7.4 and Proposition 7.3).

Thus

w = a₁b₁+ · · · + a_rb_r,

where a₁, . . . , a_r ∈ k(g) and b₁, . . . , b_r are pairwise different elements of B. Excluding, in the above equality, all the elements of the form h⁰, . . . , h^e−1 and assembling the rest of them respectively, we obtain an equality of the form (8.1) in which the elements w1, . . . , webelong to the space k(g)h₀+ · · · + k(g)h_c(s)−1= L_c(s)−1.

The uniqueness follows from the fact that the sets of the form B_sare linearly independent over k(g). 

Proposition 8.2. h^e_c(s)^s − h_c(s+1)∈ L_c(s+1)−1.

Proof. Let H = h^e_c(s)^s − h_c(s+1). If s = p, then H = h^e_c(p)^p (because h_c(p+1)= hN = 0) and then H ∈ k(f, g) = LN −1= L_c(p+1)−1. For s < p we have:

h^e_c(s)^s ∈ (A_c(s))^es ⊆ A_esc(s) = A_c(s+1)

(see Proposition 7.3) and h_c(s+1) ∈ A_c(s+1). But A_c(s+1) = f^c(s+1) + L_c(s+1)−1, so H ∈ L_c(s+1)−1. 

As a consequence of the above two propositions we get

Corollary 8.3. There exist uniquely determined elements w1, w2, . . . , we, belonging to L_c(s)−1, such that

h^e− h_c(s+1)= w₁h^e−1+ w₂h^e−2+ · · · + w_e−1h¹+ w_eh⁰, (8.2) where h = h_c(s) and e = e_s. 

Definition 8.4. The element w₁∈ L_c(s)−1 from Corollary 8.3 we will denote by w(H, s).

Proposition 8.5. deg w(H, s) < deg hc(s).

Proof. Let h = h_c(s), e = e_s, B = B_s = {h_nh^j; 0 6 n < c(s), 0 6 j < e} and let

h^e− h_c(s+1) = a1b1+ · · · + arbr, (8.3) where a₁, . . . , a_r∈ k(g) and b₁, . . . , b_r are pairwise different elements of B. Since the degrees of b₁, . . . , b_rare pairwise incongruent modulo d₀ (Lemma 7.4), there exist q ∈ {1, . . . , r} such that

deg(h^e− h_c(s+1)) = deg(aqbq) (8.4) and deg(h^e− h_c(s+1)) > deg(aibi) for all i 6= q. This implies that the number deg(h^e− h_c(s+1)) belongs to the ideal D_s. Moreover we have:

deg h^e= ed_c(s)∈ D_s and deg h_c(s+1) = d_c(s+1) 6∈ D_s.

Hence deg h_c(s+1)< deg h^e and so, deg(h^e− h_c(s+1)) = deg h^e= e deg h. Thus we have:

deg(a_qb_q) = e deg h and deg(a_ib_i) < e deg h for i 6= q. (8.5) Observe that if deg b_j 6∈ D_s−1(where j ∈ {1, . . . , r}), then j 6= q. In fact, if j = q then, by (8.5), deg(ajbj) = e deg h = esd_c(s) ∈ D_s−1 (see the definition of the number es). Moreover, deg a_j ∈ D₀ ⊆ D_s−1. Then we have a contradiction: deg b_j = deg(a_jb_j) − deg a_j ∈ D_s−1. Therefore j 6= q and, by (8.5), we know that

if deg bj 6∈ D_s−1, then deg(ajbj) < e deg h. (8.6) Let us return to the equality (8.3) and look at the elements b1, . . . , br belonging to B.

Choose of these elements all those which contain the factor h^e−1. If such elements do not exist then from the uniqueness of the presentations (8.3) and (8.2) it follows that w(H, s) = 0 and then our proposition is proved, because then deg w(H, s) = −∞ < deg h (since h 6= 0).

Assume now that {b1, . . . , bm}, where m 6 r, is the set of all such elements among b₁, . . . , b_r, which contain the factor h^e−1. Then we have:

a1b1+ · · · + ambm = w(H, s)h^e−1. (8.7) Let j ∈ {1, . . . , m}. Then bj = hnh^e−1, for some n such that 0 6 n < c(s). This implies that deg b_j 6∈ D_s−1. In fact, suppose that the number deg b_j = deg h_n+ (e − 1)d_c(s) belongs to the ideal Ds−1. Then the number (e − 1)d_c(s) also belongs to this ideal (because deg hn = dn∈ Ds−1). It is a contradiction with the property of the minimality of e.

Now we deduce, by (8.6), that deg(ajbj) < e deg h, which means (see (8.7)) that deg(w(H, s) + (e − 1) deg h = deg(w(H, s)h^e−1< e deg h,

and hence deg w(H, s) < deg hc(s). 

9 New α-systems

Let H = (h1, . . . , h_{N −1}) be an α-system. Let h₀ = g, h_N = 0 and assume that s is a fixed number from the set {1, . . . , p}. Denote by H the sequence (h1, . . . , hN −1) of rational functions belonging to the field k(f, g), which are defined as follows:

Definition 9.1.

hn=

( hn, for n 6= c(s),

h_c(s)− (1/e_s)w(H, s), for n = c(s).

Proposition 9.2. H is an α-system.

Proof. The facts that h_c(s) ∈ A_c(s) = f^c(s)+ L_c(s)−1 and w(H, s) ∈ Lc(s)−1 imply that h_c(s) ∈ A_c(s). Thus h_n ∈ A_n, for n ∈ {1, . . . , N − 1}. We know (see Proposition 8.5) that deg w(H, s) < deg hc(s). It implies that deg h_c(s) = deg h_c(s), and therefore the numbers 0, deg h₁, . . . , h_{N −1}are pairwise incongruent modulo deg g. 

In the proof of the next proposition we will use the following lemma.

Lemma 9.3. If a₂, a₃, . . . , a_e ∈ L_c(s)−1, then there exist uniquely determined elements b₂, b₃, . . . , be∈ L_c(s)−1 such that

a₂h^e−2+ a₃h^e−3+ · · · + a_e = b₂h^e−2+ b₃h^e−3+ · · · + b_e, where h = h_c(s), h = h_c(s) and e = e_s.

Proof. Let u = a₂h^e−2+ · · · + a_e. Observe that u ∈ L(e−1)c(s)−1. In fact,

u ∈ L_c(s)−1(A_c(s))^e−2 ⊆ L_c(s)−1(L_c(s))^e−2 ⊆ Lc(s)−1+(e−2)c(s)= L(e−1)c(s)−1. (9.1) The space L(e−1)c(s)−1 is obviously included in L_ec(s)−1 = L_c(s+1)−1, therefore u ∈ L_c(s+1)−1 and, by Proposition 8.1, there exist uniquely determined elements b₁, . . . , b_e ∈ L_c(s)−1 such that

u = b1h^e−1+ b2h^e−2+ · · · + be.

We must show that b₁ = 0. Suppose the contrary. Let b₁ ∈ L_nr Ln−1, where 0 6 n < c(s) (we assume that L−1= 0). Then b₁h^e−1 ∈ Ln+(e−1)c(s)r V , where V = Ln−1+(e−1)c(s) and

b₁h^e−1= u − u, (9.2)

where u = b2h^e−2+ · · · + be. In the same way as in (9.1) we show that u ∈ L(e−1)c(s)−1. But L(e−1)c(s)−1 ⊆ V . Thus the right side of the equality (9.2) belongs to V and the left side does not belong to V . The obtained contradiction implies that b1 = 0 and it ends the proof of our lemma. 

Proposition 9.4 (compare [6] page 403). Let w = w(H, s) and w = w(H, s). Assume that w 6= 0.

(1) If w ∈ L0 = k(g), then w = 0.

(2) If w ∈ L_j r Lj−1 where 0 < j < c(s), then w = 0 or w ∈ L_ir Li−1 for some i such that 0 6 i < j.

Proof. Let e = es, h = h_c(s), h = h_c(s)= h − (1/e)w and let

R =

e−2

X

r=0

e r

 h^r(−1

ew)^e−r

(recall that e > 2; see Section 7). Then

h^e− h_c(s+1)= wh^e−1+ w₂h^e−2+ · · · + w_e−1h¹+ w_eh⁰, where w2, . . . , we∈ L_c(s)−1 and

h^e− h_c(s+1) = (h − ¹_ew)^e− h_c(s+1)

= h^e− ^e₁
h^e−1(¹_e)w¹+ R − h_c(s+1)

= (h^e− h_c(s+1)) − wh^e−1+ R

= (w2h^e−2+ · · · + we−1h¹+ we) + R.

It follows from Lemma 9.3 that there exist elements b2, . . . , be∈ L_c(s)−1 such that w2h^e−2+

· · · + w_e−1h¹+ we= b2h^e−2+ · · · + be−1h¹+ be, and therefore

h^e− h_c(s+1)= (b₂h^e−2+ · · · + b_e−1h¹+ b_e) + R. (9.3) Now we will deal with the component R. Assume that w ∈ L_j, where 0 6 j < c(s) and let m ∈ {0, 1, . . . , e − 2}. Then

h^mw^e−m∈ L^m_c(s)L^e−m_j ⊆ Lmc(s)+j(e−m)

and it is easy to show that mc(s) + j(e − m) 6 (e − 1)c(s) + j − 1. This implies that

R ∈ L(e−1)c(s)+j−1. (9.4)

Observe that (e − 1)c(s) + j − 1 6 ec(s) − 1 = c(s + 1) − 1. Thus R ∈ Lc(s+1)−1 and so, by Proposition 8.1, there exist uniquely determined elements z1, . . . , ze∈ L_c(s)−1 such that

R = z1h^e−1+ · · · + ze−1h¹+ ze. (9.5) Now from (9.3) and (9.5) we obtain the equality

w = w(H, s) = z1. Observe also that

z₂h^e−2+ · · · + z_e∈ L_c(s)−1(L_c(s))^e−2⊆ L(e−1)c(s)−1. Thus, by (9.4), we have:

wh^e−1= z₁h^e−1 ∈ L(e−1)c(s)+j−1. (9.6) Now it is not difficult to prove the properties (1) and (2). Assume that w 6= 0. Then w ∈ L_ir Li−1for some i such that 0 6 i < c(s), assuming that L−1= 0. Since h^e−1 ∈ A_(e−1)c(s), then

wh^e−1 ∈ L(e−1)c(s)+ir L(e−1)c(s)+i−1. (9.7) If j = 0, then the property (9.6) contradicts with the property (9.7). Thus, if j = 0 (that is, if w ∈ k(g)), then w = 0. Therefore we have (1).

Assume now that j > 0. Then, by (9.6) and (9.7), we get (e − 1)c(s) + i − 1 < (e − 1)c(s) + j − 1, and so i < j. Therefore we have (2). 

Starting from an arbitrary α-system we may construct a new α-system H. By the same way, an α-system H emerges of an α-system H, etc. We can make such a conversion suffi- ciently many times (for each s = 1, . . . , p). Then, by Proposition 9.4, we obtain:

Proposition 9.5. There exists an α-system H such that w(H, s) = 0, for each s ∈ {1, . . . , p}.



Note the following lemma.

Lemma 9.6 ([2]). Let e, n be natural numbers such that en 6 N . Let u, v ∈ k(f, g).

Assume that u ∈ A_n, v ∈ A_enand u^e− v ∈ L_en−n−1 (assuming that L−1 = 0). Then u ∈ A_n. A proof of this lemma we will present in the last section in which we will prove a more general fact (Proposition 11.1). Now, using the above lemma, we will prove the following proposition.

Proposition 9.7. There exists an α-system H = (h1, . . . , hN −1) such that each element of the form h_c(s) (dla s = 1, . . . , p) belongs to the ring k[f, g].

Proof. Let H be an α-system such as in Proposition 9.5. We will show that if s ∈ {1, . . . , p}, then h_c(s) ∈ k[f, g].

Assume first that s = p. Let u = h_c(p), v = h_c(p+1) = h_N = 0, e = e_p, n = c(p). Then u, v ∈ k(f, g), en = epc(p) = c(p + 1) = N , u ∈ An and the equality N = (k(g)(f ) : k(g)) implies that v = 0 ∈ AN = Aen. Moreover (by Corollary 8.3 and Definition 8.4),

u^e− v = h^e_c(p)− h_c(p+1)= w₂h^e−2_c(p)+ · · · + w_e−1h_c(p)+ w_e, where w₂, . . . , w_e are some elements belonging to L_c(p)−1. This implies that

u^e− v ∈ L_c(p)−1(L_c(p))^e−2 ⊆ Lec(p)−c(p)−1= L_en−n−1.

Therefore all the assumptions of Lemma 9.6 are satisfied. Thus h_c(p)= u ∈ An⊆ k[f, g].

Now let s + 1 6 p and assume that hc(s+1) ∈ k[f, g]. Denote: u = h_c(s), v = h_c(s+1), e = es

and n = c(s). Then en = e_sc(s) = c(s + 1) 6 c(p) < N , u ∈ An and v ∈ k[f, g] ∩ A_c(s+1) = A_c(s+1)= A_en(see Proposition 2.1). As above (thanks to the fact that w(H, s) = 0) we show that u^e− v ∈ L_{c(s)es−c(s)−1} = Len−n−1. Thus all the assumptions of Lemma 9.6 are again satisfied. Therefore h_c(s) = u ∈ A_n⊆ k[f, g]. 

10 The existence of β-systems

Now we are ready to prove the theorem on the existence of a β-system announced before.

The proof of Theorem 3.4. Let H = (h1, . . . , h_{N −1}) be an α-system such as in Propo- sition 9.7. Let

H⁰ = {h^j_c(1)¹ · · · h^j_c(p)^p ; 0 6 js < es for s = 1, . . . , p}, J = {j1c(1) + · · · + jpc(p); 0 6 js< es for s = 1, . . . , p}.

The fact that h_c(1), . . . , h_c(p)∈ k[f, g], implies that H⁰ ⊂ k[f, g]. Each element of the form h^j_c(1)¹ · · · h^j_c(p)^p belongs to the set A_{j1c(1)+···+jpc(p)}. We will show that the elements of the set H⁰ one can arrange in a sequence (h⁰₀ = 1, h⁰₁, . . . , h⁰_{N −1}) which will be a β-system. For this purpose it is enough to prove the following four properties:

(1) Each element of J is smaller than N . (2) The elements of J are pairwise different.

(3) The cardinality of J equals N .

(4) The degrees of polynomials belonging to H⁰ are pairwise incongruent modulo deg g.

Proof (1). We know (see Proposition 7.3) that e_sc(s) = c(s + 1), for s = 1, . . . , p. We also know that c(p + 1) = N . Thus we have:

j₁c(1) + · · · + j_pc(p) < e₁c(1) + j₂c(2) + · · · + j_pc(p)

= c(2) + j₂c(2) + · · · + j_pc(p) 6 e2c(2) + j₃c(3) + · · · + j_pc(p)

= c(3) + j3c(3) + · · · + jpc(p) ...

6 epc(p) = c(p + 1) = N.

Proof (2). Let

j₁c(1) + · · · + j_pc(p) = i₁c(1) + · · · + i_pc(p), (10.1) where i₁, j₁< e₁, . . . , i_p, j_p < e_p. Since c(1) = 1, c(2) = e₁, c(3) = e₁e₂, . . . , c(p) = e₁· · · e_p−1 (see Corollary 7.7) then, by the equality (10.1), the number |i₁ − j₁| is divisible by e₁, and thus i1= j1 (because i1, j1< e1). Now we have the equality

j₂c(2) + · · · + j_pc(p) = i₂c(2) + · · · + i_pc(p)

from which it follows (after the division by e₁) that the number |i₂ − j₂| is divisible by e₂, that means i2 = j2. Repeating this procedure we obtain that is= js for all s = 1, . . . , p.

Proof (3). From (2) it follows that the set J contains exactly e1· · · e_p= N elements.

Proof (4). Let

a = hⁱ_c(1)¹ · · · hⁱ_c(p)^p , b = h^j_c(1)¹ · · · h^j_c(p)^p ,

where 0 6 is, js< ee, for s = 1, . . . , p, and assume that deg a ≡ deg b (mod deg g). Then (i1d_c(1)+ · · · + ipd_c(p)) − (j1d_c(1)+ · · · + jpd_c(p)) ∈ (d0),

and thus |i_p− j_p|d_c(p)∈ D_p−1and, from the property of the minimality of e_p, we obtain the equality i_p = j_p. In a similar way we deduce that i_p−1= j_p−1. Repeating this procedure we can see that a = b. It ends the proof of the property (4) and the proof of Theorem 3.4. 

11 The proof of Lemma 9.6

It has remained to prove Lemma 9.6. We will do it in the present section. For this purpose first we will prove the following proposition.

Proposition 11.1. Let R ⊂ P be commutative rings containing the field Q of rational num- bers. Let F ∈ P [x] be a monic polynomial of the degree n > 1 and let e > 1 be a natural number. Consider the polynomial

F^e = x^ne+ bne−1x^ne−1+ bne−2x^ne−2+ · · · + b0. If bne−1, bne−2, . . . , bne−n ∈ R, then F ∈ R[x].

For the proof of this proposition we need certain lemmas. Let n, e be positive integers and let

L(n, e) = {(in, . . . , i0) ∈ Zⁿ⁺¹; in+ · · · + i0= e, is> 0 for s = 0, . . . , n}.

Consider the following partial order 6 on the set L(n, e):

(i_n, . . . , i₀) 6 (jn, . . . , j₀) ⇐⇒











in 6 jn

in−1+ in 6 jn+ jn−1

...

i0+ · · · + in−1+ in 6 jn+ jn−1+ · · · + j0. Consider also the function ϕ : L(n, e) −→ N ∪ {0} defined by the formula:

ϕ(in, . . . , i0) = nin+ (n − 1)in−1+ · · · + 1i1+ 0i0. Note the following obvious lemma.

Lemma 11.2. Let i, j ∈ L(n, e). Then:

(1) i 6 j ⇒ ϕ(i) 6 ϕ(j);

(2) i < j ⇒ ϕ(i) < ϕ(j);

(3) If i 6= j and ϕ(i) = ϕ(j), then i, j are not comparable with respect to 6.  Let πn−1, . . . , π0 be elements of L(n, e) defined as follows:

π_n−1 = (e − 1, 1, 0, 0, . . . , 0, 0), πn−2 = (e − 1, 0, 1, 0, . . . , 0, 0), ...

π1 = (e − 1, 0, 0, 0, . . . , 1, 0), π₀ = (e − 1, 0, 0, 0, . . . , 0, 1).

The next lemma is also obvious.

Lemma 11.3. Let i = (in, . . . , i0) ∈ L(n, e) and let s ∈ {0, 1, . . . , n}. Then:

(1) if i_s> 1, then ϕ(π_s) > ϕ(i);

(2) if i_r> 0 for some r < s, then ϕ(π_s) > ϕ(i). 

Proof of Proposition 11.1. Let F = xⁿ+ a_n−1xⁿ⁻¹+ · · · + a₁x + a₀, where a_n−1, . . . , a₀ ∈ P . Then:

F^e= X

in+···+i0=e

hi_n, . . . , i0iaⁱ_n−1ⁿ⁻¹· · · a₀ⁱ⁰x^{ϕ(in,...,i0)}, (11.1) where hi_,. . . , i₀i = e!(i_n! · · · i₀!)⁻¹. We will show, using an induction, that the elements a_n−1, . . . , a0 belong to R.

First we prove that an−1 ∈ R. For this purpose study the coefficient standing by x^ne−1 in (11.1). Since ne − 1 = ϕ(π_n−1), we have in (11.1) the component

he − 1, 1, 0, . . . , 0ia¹_n−1x^ne−1.

Let i = (in, . . . , i0) be such an element of L(n, e) which satisfies the equality ϕ(i) = ne − 1 = ϕ(πn−1).