Decidability of the Theory of Addition and the Frobenius Map in Rings Of

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Theorem 1.1 Let F be an algebraic ﬁeld extension of Fp. Let R be as above. Then the following hold:

e 1. The Lp(z) -theory of R is eﬀectively model-complete.

e 2. Every Lp(z) -sentence is equivalent in R to a sentence of the form Pσ, where σ is a sentence of Lp.

We prove model-completeness by constructing an algorithm which converts any exis- e e tential Lp(z) -formula to an equivalent, in R, universal Lp(z) -formula. It is well known that model completeness of the theory of a countable model with a recursive elementary diagram implies decidability of the theory (see current developments in [CMS21]), so we obtain:

e Corollary 1.2 Assume that the Lp-theory of the field F is decidable. Then the Lp(z) - theory of R is decidable. Item 2 of Theorem 1.1 says that ‘questions’ (first-order sentences) about R may be effectively translated into questions about F (as a model of Lp). This replies positively, for the structures R, to a ‘program’ asked by Leonard Lipshitz: ‘Identify theories with universe a polynomial ring F [z] or a field of rational functions, extending the structure of addition by commonly used operations and relations, which have the property that one can effectively translate first-order sentences of the structure into questions about F and, possibly, other elementary mathematical structures, e.g., groups’. This follows the type of results of J. Ax and S. Kochen for fields of p-adic numbers in [AK65]. Theorem 1.1 is, in part, a generalization of the results of [PZ04], where a similar theorem was proved for rings of polynomials (i.e., when S is the empty set), for any perfect field F (not necessarily algebraic). A similar result (model-completeness) was proved in [Ona18] by Onay for the henselization of Fp[z], seen as a module over Fp[z] (and, more generally, over a finite field or an algebraic closure of it). The problem e of whether the Lp(z) -theory of the field Fp(z) or F˜p(z) is model-complete, or even decidable, remains open. The structure of addition and the Frobenius map is interesting, not only for its own sake, but also because it is connected to various important algebraic and logical

2 problems. For example, the derivative of a function (polynomial, rational or power e series) is positive-existentially definable in Lp(z) (see the Introduction of [PZ04]). So, the structure of R as a model of addition and differentiation is encodable in its Lp(z)- structure. For surveys regarding the decidability properties of algebraic structures the reader may consult [PZ00], [PZ08], [Poo08] and [Koe18]. For other decidability results for polynomial rings of positive characteristic the reader may consult [Sir10] - an analogue of results of A. Semenov for the natural numbers with addition and the set of powers of a fixed prime number - and [Phe85] where it is proved that the existential theory of addition and divisibility over a ring of polynomials with coefficients in an existentially decidable field is decidable. For the algebraic and model theoretic properties of the Frobenius map see [Maz75], [CH99] and [Hru04]. Although the general strategy of proof of [PZ04] works for the rings R, several of its components are quite difficult to adapt. We describe briefly the main two. A m polynomial f ∈ Fp[z], in m variables, is additive if, for all a and b in (F˜(z)) , it satisfies f(a)+f(b)= f(a+b). Notice that the polynomial terms of the language Lp(z), with a zero constant term, are such additive polynomials. Here we will consider only additive polynomials with coefficients in Fp[z]. Such an f is called strongly normalized if its coefficients are in Fp[z], the degrees of f with respect to each of its variables is the same, ps, for some s ∈ N ∪{0} and the degrees of its leading coefficients are pairwise inequivalent modulo ps. We develop an algorithm by which questions regarding solvability of - arbitrary - additive polynomials are reduced to similar questions for strongly normalized polynomials. The first crucial property of a strongly normalized polynomial f is that, for any given y ∈ R, the inverse image {x ∈ Rm : f(x)= u} has a bounded height. This is relatively easy for the case that R is a polynomial ring and we know this is not true if R is substituted by the field of all rational functions Fp(z). In order to prove it for the rings R we use the notion of a ‘Hasse derivative’ and recent results on the relative Algebra (see Section 3). So we prove:

Theorem 1.3 Let f be a strongly normalized additive polynomial of the variables x = (x1,...,xn). Then there is a recursive function h which, to each additive polynomial f of the language Lp(z) and each ℓ ∈ N associates a non-negative integer h(f,ℓ) such that the height of each element of the set {x ∈ F (z)n : |f(x)| ≤ ℓ} is less than or equal to h(f,ℓ). A second point where the strategy of [PZ04] needs significant adaptations is where we need to prove that the image of a strongly normalized polynomial, whose leading s coefficients form a basis of F (z) over F (zp ) (where ps is the degree of the polynomial), is ‘almost all of R’ (in the sense of Lemma 2.4). Our method works only for algebraic fields F - not for general perfect fields. Item 2 of Theorem 1.1, even for the polynomial case, is obviously stronger than the results of [PZ04] and its proof requires some improvements in the logical treatment of the subject - see the proof of Section 4. In the rest of this section, we fix our notation and give a sketch of the proof.

3 Section 2 contains some necessary algebraic results. We prove Theorem 1.3 in Section 3. and Theorem 1.1 in Section 4. Some tedious elementary proofs are gathered in the Appendix.

1.1 Notation, Deﬁnitions and some elementary algebraic Facts We use the following notations and deﬁnitions.

1. N is the set of natural numbers.

2. p is a prime number, Fp is a ﬁeld with p elements, F˜p is an algebraic closure of Fp.

3. F is a ﬁeld of characteristic p such that Fp ⊆ F ⊆ F˜p.

4. z is a variable, F [z] is the ring of polynomials of z with coefficients in F and F (z) is its field of quotients, i.e., rational functions of z with coefficients in F .

5. S is a ﬁnite set S = {s1,...,sν} of irreducible elements of Fp[z], which we assume to remain irreducible as elements of F [z].

−1 −1 6. R is the ring F [z,s1 ,...,sν ], i.e., the ring generated over F [z] by the inverses of the elements of S.

p 7. The language Lp is deﬁned as Lp := {+, x 7→ x , =, 0, 1}. Its symbols are interpreted as follows: + denotes addition, x 7→ xp denotes the Frobenius map, = denotes equality, 0 and 1 are constant symbols for the obvious elements of Fp.

8. The language Lp(z) is deﬁned by Lp(z) = Lp ∪ {∈ F, x 7→ zx} where x 7→ zx is the multiplication-by-z map, and ∈ F is a unary predicate denoting belonging to F .

9. If α =(α1,...,αm) is a tuple of the variables αi, then α ∈ F denotes the relation m ∧i=1αi ∈ F .

e 10. The language Lp(z) is the extension of Lp(z) by the predicates Pσ, where the index σ ranges over the set of formulas σ(a1,...,an) of Lp. We assume that all the free variables of σ are among a1,...,an. The predicate Pσ associated to σ(a1,...,an) is interpreted as the n-ary relation “a1,...,an ∈ F and σ(a1,...,an) is true in F ”.

e 11. R denotes the model of the language Lp(z) , with universe R, with symbols interpreted as above.

12. A variable ranging only over F will be called an F -variable.

4 13. For x ∈ F (z), if Q is an irreducible element of F [z], we write ordQ(x) for the order of x at Q. Recall that by convention ordQ(0) = ∞. If Q = z − ρ, with ρ ∈ F , then we write ordρ instead of ordQ. The order at inﬁnity of x, denoted ord∞(x), is the degree of the denominator of x minus the degree of its numerator. 14. An additive polynomial is a polynomial of the form n

f(x)= fi(xi), (1) i=1 X where x =(x1,...,xn) and, for each i,

s(i) ps(i) ps(i)−j fi(xi)= bixi + ci,jxi , (2) j=1 X s(i) with bi,ci,j ∈ Fp[z]. The degree of f is deg(f) := max{p }. i 15. We say that the additive polynomial f(x) is a polynomial of all the variables of the tuple x =(x1,...,xm) whenever the degree of f in each of the variables xi is positive, for i =1,...,m.

ps 16. For s ∈ N, let Vs(Fp) be Fp(z) considered as a vector space over the field Fp(z ). ps Respectively, let Vs(F ) be F (z) considered as a vector space over the field F (z ). 17. An additive polynomial f is called normalized if all degrees s(i) are equal and the set of leading coefficients {bi : i = 1,...,n} is linearly independent over Vs(Fp). An additive polynomial is called p-basic if it is normalized and the set {b1,...,bn} forms a basis for Vs(Fp). Moreover, f is strongly normalized if f is s normalized and the degrees of the bi are pair-wise inequivalent modulo p , where ps is the degree of f.

18. Deﬁnition 1.4 A proper transformation is a tuple ξ =(ξ1,... ,ξn) such that each ξi(X, β) is an additive polynomial of the variables x1,...,xm, β1,...,βµ and such that the map ξ deﬁned by ξ: Rm × F µ → Rn (x1,...,xm, β1,...,βµ) 7→ (ξ1(x1,...,xm, β1,...,βµ),... , ξn(x1,...,xm, β1,...,βµ)) is surjective. Also note that the composition of proper transformations is again a proper transformation. We will user proper transformations in order to change variables. 19. Let x be an tuple of variables and f be an additive polynomial of the variables of x. When all the variables of the additive polynomial H are F -variables then we write ImF (H) := {y ∈ R | ∃α ∈ F H(α)= y}. We write Im(f) := {y ∈ R | ∃x ∈ R f(x)= y}.

5 1 F 20. A bounded term is any expression of the form e G(α), where e ∈ p[z] and G(α) is an additive polynomial of the tuple of F -variables α. Notice that a bounded e term is not always a term of the language Lp(z) but we will be writing them with the understanding that ‘clearing of the denominators’ is performed immediately e - this makes sense because the only relation symbol of Lp(z) , apart from ∈ F , is 1 1 equality. We write ImF ( e G) for e ImF (G). a 21. The height of a rational function u = b ∈ F (z), where a, b ∈ F [z] are coprime, is |u| := max{deg(a), deg(b)}. The height of 0 is not deﬁned.

22. The partial fraction decomposition of a rational function, from elementary algebra, is given by:

Fact 1.5 Let x ∈ F (z) and let Q1,...,Qr be all the monic, irreducible polynomials of F [z] at which x has a pole. Then x can be written in a unique way as

di,j(z) x = g(z)+ j , (3) i,j Qi X where the index i ranges in the set {1,...,r} and, for each i, the index j ranges in a non-empty subset of N. Each of g(z) and di,j(z) is in F [z] and for each (i, j) the degree of the polynomial di,j(z) is less than the degree of Qi.

Fact 1.6 (a) Let u ∈ R and c ∈ F [z] with deg(c) = d > 0. Then there is a v ∈ R and a polynomial r, of degree less than d, such that u = vc + r. If no irreducible factor (over F [z]) of c is invertible in R, then v and r are unique. (b) Given u ∈ R, c ∈ F [z] \{0} and N ∈ N, there are v ∈ R and ri ∈ F [z], for i =0,...,N, with deg(ri) < deg(c) such that

N N+1 u = r0 + r1c + ··· + rN c + vc .

(c) Let Q ∈ Fp[z] which is not divisible by any polynomial that is irreducible in F [z] and invertible in R, of degree d ≥ 1. Then, for any x ∈ R, the formula x 6∈ F is equivalent in R to

∃y∃α0 ... ∃αd−1 (4) d−1 d−1 [α0,...,αd−1 ∈ F ∧ x = yQ + αd−1z + ...α1z + α0 ∧ (y =06 ∨ αi =6 0)] i=1 _ The proof is elementary. For completeness, we include one in Section A.1.

Fact 1.7 A system of equations may substituted by one equation due to the equivalence (x =0 ∧ y = 0) ↔ xp + zyp =0.

6 Fact 1.8 Let F be a perfect ﬁeld. Let s ∈ N ∪{0}. Then:

i s • The set {z | 0 ≤ i ≤ p − 1} is a basis of both Vs(Fp) and Vs(F ).

• Let B be a subset of Fp(z) which is linearly independent in Vs(Fp). Then it is also linearly independent in Vs(F ). Consequently, any basis of Vs(Fp) is also a basis of Vs(F ).

1.2 Existential Formulas

e Let u be a term of Lp(z) . From Fact 1.6, the formula u∈ / F can be substituted by r−1 the equivalent ∃α, x[α ∈ F ∧ u = α0 + α1z + ··· + αr−1z + gx ∧ x =6 0], where g is an irreducible polynomial in F [z], of degree r, which is not in S. If u is a term but not a variable, we can replace the formula u ∈ F by the equivalent ∃α[α ∈ F ∧ u = α]. Therefore, any formula is equivalent in R to a formula in which the negation of the predicate-symbol ∈ F does not occur and in which ∈ F is applied only to variables. e Such a formula of Lp(z) , where all quantiﬁed variables range over F will be called a bounded formula (cf. [PZ04], p. 1021).

Fact 1.9 The set {x ∈ R\{0} : |x| ≤ k}∪{0} is deﬁnable in Lp(z) by a bounded existential formula, i.e., one which is existential and its quantiﬁed variables are all F -variables.

e We leave to the reader to verify that an existential formula of Lp(z) is equivalent in R to a disjunction of formulas of the form:

φ(u, {vj}j∈J ): ∃x, α [α ∈ F ∧ ψ(x, α)] , (5) where ψ(x, α): f(x)+ H(α)= u ∧j∈J ej(x)+ Gj(α) =6 vj ∧ Pσ(α) . (6) under the conventions:

• x =(x1,...,xm) is a tuple of the variables xi. • α is a tuple of F -variables, each of them distinct from each variable of x.

• f is an additive polynomial of all the variables of x (hence, by convention, we F have degxi (f) > 0 for every i), with coeﬃcients in p[z].

• Each ej is an additive polynomial of some of the variables of x. • H is an additive polynomial in some of the variables of α.

• Each Gj is an additive polynomial in some of the variables of α.

• u and the vj are terms of Lp(z).

7 • No variables among those of x or α occurs in u or any of the vj.

• The predicate symbol Pσ(α) may have more variables than those of α occurring in it.

1.3 Overview of this article. We start with a formula φ as above. In Section 2, we show that, through a certain type of change of variables (see proper transformation in Deﬁnition 1.4), one may assume that the polynomial f is strongly normalized or the zero polynomial. The way to do this is similar to that of [PZ04], but several details have to be adjusted to the new rings R. In particular, Lemma 2.4 constitutes a new approach. In Section 3, we prove Theorem 1.3. This allow us to substitute sets of the form {x ∈ F (z): |f(x)| ≤ k} by sets of the form {x ∈ F (z): |x| ≤ h}, where k, h ∈ N. Finally, in Section 4, we use the previous steps in order to show that φ is equivalent to a universal formula.

2 Strongly normalized polynomials and properties

The following lemmas will be used in the proof of Theorem 1.3, In this Section, we prove Theorem 1.3.

Lemma 2.1 (counterpart of [PZ04], Lemmas 3.3 and 3.4) Let f be an additive polynomial in m0 variables, with coeﬃcients in Fp[z]. Then, there is a proper transformation ξ : Rm ×F k → Rm0 , a strongly normalized additive polynomial f˜ in m variables, with coeﬃcients in Fp[z] and an additive polynomial G in only F -variables, each one of them distinct from the variables of f˜, such that:

• f ◦ ξ = f˜+ G,

• m ≤ m0,

• deg(f˜) ≤ deg(f),

• Im(f)= Im(f˜)+ ImF (G),

• ξ and f˜ are eﬀectively computable from f.

The proof is similar to those of Lemma 3.3 and 3.4 of [PZ04]. For completeness, we provide outlines in Section A.4.

Lemma 2.2 For any strongly normalized additive polynomial f˜ in the variables of x = (x1,...,xm), ranging over R, there is an additive polynomial h, in the variables of v = (v1,...,vps−m), also ranging over R, each one distinct from the variables of x, such that f˜+ h is p-basic and strongly normalized.

8 For a proof, see Lemma 3.5(a) of [PZ04].

Deﬁnition 2.3 Let f an additive polynomial over F (z), in m variables. We deﬁne the relation ∼f over F (z) by

′ ′ m u ∼f u if and only if u − u = f(˜x), for some x˜ ∈ R . (7)

Lemma 2.4 Assume that F ⊂ F˜p. Then there is a primitive recursive function Eord from the set of p-basic polynomials of Lp(z) to N such that, for any p-basic polynomial f over Fp[z] the following hold: 1. For any u ∈ R, there exists u′ ∈ R with the following properties:

′ • u ∼f u . • Each pole of u′ is a pole of u - the word ‘pole’ includes the pole at inﬁnity. ′ • For all ρ ∈ F˜p ∪ {∞}, we have |ordρ(u )| ≤ Eord(f).

s 2. Assume that f has degree p , with s ≥ 0. Write N = Eord(f), let α = (α0,..., N−1 i αN−1) be a tuple of distinct F -variables and G(α) := i=0 αiz . Then, 1 P R = Im(f)+ Im ( G), (8) F eN where e is the product of all elements of S. Proof : Consider an arbitrary s ≥ 0, write q = ps and consider a p-basic polynomial q q F f(x) = i=1 bixi + g(x), of degree q, with coeﬃcients in p[z], where x =(x1,...,xq) is the tuple of variables of f and g is an additive polynomial in x, of degree less than q, henceP at most q/p. i Since f is p-basic, both {z : 0 ≤ i ≤ q − 1} and {b1,...,bq} are bases of Vs(Fp). Therefore, there is a non-singular matrix A such that B = AZ, where Z and B are i the vectors of the z and the bi respectively - in some order. Obviously A has entries q in Fp[z ]. Write ∆ for the determinant of A. Using Cramer’s Rule to solve the system B = AZ for the variable zi, we obtain that, for any i ∈ {0,...,q − 1}, there are ei,j ∈ Fp[z] such that q i q ∆z = ei,jbj. (9) j=1 X We will now prove this auxiliary claim.

a N Claim 2.5 Let v = Qℓ , where a, Q ∈ F [z], ℓ ∈ and Q is a monic, irreducible polynomial over F [z]. If ∆ divides a then there exists a v′ ∈ F [z] such that

′ 1. v ∼f v , 2. the poles of v′ are roots of Q and,

9 ′ ℓ+r 1 3. |ordQ(v )| ≤ p = p (|ordQ(v)| + r), where r is the least non-negative integer such that p|ℓ + r.

Proof : Let r be the least non-negative integer such that ℓ + r = kq for some k ∈ N. Clearly 0 ≤ r < q. Since ∆ divides a, we write aQr = ∆a′ for some a′ ∈ F [z]. By Fact 1.8, there are ai ∈ F [z] so that ps−1 ′ q i a = ai z . i=0 X i Then, using 9, there are ei,j ∈ F [z] such that, for each i = 0,...q − 1 we have ∆z = q q j=1 ei,jbj, hence

P q−1 q i q−1 q q q q a ∆z a e bj v = i=0 i = i=0 j=1 i i,j = b x˜q, Qkq Qkq j j P P j=1 P X 1 q−1 wherex ˜j := Qk i=0 aiei,j. Settingx ˜ := (˜x1,..., x˜q), we have v = f(˜x) − g(˜x). Clearly, everyx ˜j is in R, since Q is an element of S hence invertible in R. Furthermore, observe P q F that the order of g(˜x) at Q is at least −k p , since the coefficients of g are in p[z]. So we set v′ = −g(˜x) and the proof is complete. ✸ We now proceed in the proof of Item 1 of the Lemma. Consider any u ∈ F (z). a Write it as partial fractions, i.e., as sum of terms of the form uQ,ℓ = Qℓ or h, where a, h, Q ∈ F [z], ℓ ∈ N and Q is a monic, irreducible polynomial over F [z]. For each term ′ uQ,ℓ, we will find a uQ,ℓ ∈ F (z), whose poles (except the one at infinity) are roots of Q, with order bounded by Eord, to be determined, such that

′ uQ,ℓ ∼f uQ,ℓ.

Obviously ∼f is an equivalence and additive relation, so u will be related by ∼f to the ′ sumu ˆ = Q,ℓ uQ,ℓ + h. Finally, we will treat the polynomial part h ofu ˆ in a similar way, to obtain a u′ with the required properties. P Since ∆ is in Fp[z], there is a ﬁnite extension Fpm of Fp in which ∆, as a polynomial in z, splits into linear factors (and m ≥ 1). Then, the product of the distinct irreducible m p m0 (over Fp[z]) factors of ∆ divides z − z, in Fp[z]. Taking some high enough power p of p, such that the multiplicity of each irreducible factors of ∆ is ≤ pm0 we have that m m m m m ∆ divides (zp − z)p 0 = zp + 0 − zp 0 . We rewrite the term uQ,ℓ as

m m m m a a(Qp + 0 − Qp 0 ) aQp u = = − + . (10) Q,ℓ Qℓ Qℓ+pm0 Qℓ

m+m0 m0 m a(Qp −Qp ) aQp pm m Let u1 := − Qℓ+p 0 and u2 := Qℓ = Q uQ,ℓ. Since m> 0, we have that the order of u2 at Q is greater than the order of uQ,ℓ. Moreover, the numerator of u1 is

10 divisible by ∆, based on the remarks of the previous paragraph. Hence, by Claim 2.5, ′ 1 ′ ′ 1 m0 there is a u1 ∈ F [z, Q ] such that u1 ∼f u1 such that |ordQ(u1)| ≤ p (ℓ + p + r), where r is the least non-negative integer such that p|ℓ + pm0 + r. m0 1 m0 m0 p +q Note that if p (ℓ + p + r) ≥ ℓ then p + r ≥ (p − 1)ℓ and therefore, ℓ < p−1 , pm0 +q ′ since r < q. Consequently, if |ordQ(u1)| = ℓ ≥ p−1 then ordQ(u1) < −ℓ = ordQ(u1). ′ ′ ′ In this case, we have uQ,ℓ ∼f uQ,ℓ := u1 + u2 and |ordQ(uQ,ℓ)| < |ordQ(uQ,ℓ)|. After iterating this procedure for all terms uQ,ℓ of u, and adding the equivalency relations, we will end up with

′ ′ u = uQ,ℓ + h ∼f u := uQ,ℓ + h (11) Q,ℓ Q,ℓ X X Applying the partial fractions decomposition to u′, we obtain

′ u = uˆQ,ℓ + hˆ + h, (12) Q,ℓ X whereu ˆQ,ℓ ∈ F (z) and hˆ ∈ F [z]. Using Lemma 3.5(d) of [PZ04] for the polynomial

ω := hˆ + h, there is aω ˆ ∈ F [z] such that ω ∼f ωˆ and |ord∞(ˆω)| ≤ Ω, where Ω is a bound depending only on f. Deﬁne Eord to be

pm0 + q E (f) := max{ , Ω}, (13) ord p − 1

and Item 1 of the Lemma is proved. Item 2 follows clearly. ✸ Note: As seen by its proof, Lemma 2.4 is true for any subring of F˜p(z) in the place of R. Moreover, observe that the condition F ⊂ F˜p seems to be essential.

3 The order of poles of strongly normalized polynomials is bounded

In this Section, we prove Theorem 1.3. For the rest of this Section, F is an algebraically closed ﬁeld of characteristic p> 0.

3.1 Hasse derivatives and properties. Hasse derivative or hyperderivative [Has36; Sch76] is a generalization of the derivative for ﬁelds of rational functions. It is especially useful in positive characteristic. The ǫ − th Hasse derivative of zj is deﬁned as

j D (zj) := zj−ǫ, for j ≥ 0. (14) ǫ ǫ 11 For ǫ = 0, D0 is the identity function and we write D instead of D1. In zero characteristic, it holds that 1 d D (zj)= zj . (15) ǫ ǫ! dz A set of useful properties is provided in [Jeo11]. Some of these properties are - f,g are rational functions of the variable z over a ﬁeld F :

P1 The hyperderivative is linear, i.e.,

Dǫ(f + g)= Dǫ(f)+ Dǫ(g), (16)

P2 The hyperderivative satisﬁes the Leibniz product formula:

Dǫ(fg)= Di(f)Dj(g), (17) i+j=ǫ X where f,g ∈ F (z) and i, j, ǫ ≥ 0.

P3 If p> 0 is the characteristic of F , m, ǫ ∈ N and f ∈ F (z), then

pm m pm (Dj(f)) if ǫ = jp , Dǫ(f )= m (18) (0 if ǫ 6≡ 0 (mod p ).

P4 For ǫ ∈ N and 0 =6 f ∈ F (z),

ǫ 1 (−1)j D = D (f) ··· D (f). (19) ǫ f f j+1 i1 ij j=1 i1,...,ij ≥1 X i1+X···+ij =ǫ

We will apply the above for a ﬁeld F of characteristic p. Let q = ps for some s ∈ N. Let ǫ ∈ {1, 2,...,ps − 1}. Since ǫ 6≡ 0 mod ps, we conclude that, for every f ∈ F (z), we have ps Dǫ(f )=0, ∀ǫ ∈{1,...,q − 1}. (20)

q Let B = {β1,...,βq} be a basis of F (z) over F (z ) then, for every g ∈ F (z), there exist g1,...,gq ∈ F (z) such that

q q g = g1β1 + ··· + gq βq. (21)

Then due to 16, 17 and 20, it holds that

q q Dǫ(g)= g1Dǫ(β1)+ ··· + gq Dǫ(βq), ∀ǫ ∈{1,...,q − 1}. (22)

12 3.2 Linear independence criterion in positive characteristic Now, we present a theorem between Wronskians and linearly independent sets of functions similar to the well known theorem of Calculus. The corresponding theorem regarding the linear independence and the respective Wronskian (with hyperderivatives instead of classic ones), for ﬁelds of positive characteristic, was initially described in [GV87] and strengthened in [Wan99]. Recall that we work over F which is a ﬁeld of characteristic p> 0.

Theorem 3.1 [Th.1, [GV87]] Let x1,...,xn ∈ F (z). Then x1,...,xn are linearly s independent over Vs(F ) if and only if there exist integers 0 = ǫ1 < ǫ2 < ··· < ǫn < p with det (Dǫi (xj)) =06 . ps Since Vs(F ) is F (z), seen as a vector space over F (z ), it follows that:

Corollary 3.2 The set {b1, b2,...,bn} ⊂ F (z) is linearly independent over Vs(F ) if s and only if there exist integers 0= ǫ1 < ǫ2 < ··· < ǫn

3.3 The orders of hyperderivatives

Let u ∈ F (z)\{0}. Fix λ ∈ F and drop the subscript in ordλ(u) for convenience. We will give an upper bound of ord(Dǫ(u)). Initially, we consider the base {1, z − λ, (z − λ)2,..., (z − λ)q−1} of F (z) over F (zq) and rewrite u as

q q 2 q q−1 q u = u0 +(z − λ)u1 +(z − λ) u2 + ··· +(z − λ) uq−1, (23) for some u0,u1,...,uq−1 ∈ F (z). Next, we apply the ǫ-th hyperderivative, for ǫ ∈ {1,...,q − 1}, and deduce that

q q 1 q q−1 Dǫ(u)= u0Dǫ(1) + u1Dǫ (z − λ) + ··· + uq−1Dǫ (z − λ) . (24) It is now easy to see, using the composition properties of [Jeo11], that

k D (z − λ)k = (z − λ)k−ǫ. (25) ǫ ǫ k Observe that Dǫ (z − λ) vanishes if k < ǫ. Since ord c(z − λ)k = k, when k ∈ N and c is a non-zero constant, for k ≥ ǫ, Equation 25, is equivalent to k ord Dǫ (z − λ) = k − ǫ. (26)

We are now ready to prove the following.

Theorem 3.3 Let u ∈ F (z)\{0}, where F is a algebraically closed ﬁeld of positive characteristic. If λ ∈ F , ǫ ∈ N and u, Dǫ(u) =06 , then ordλ (Dǫ(u)) ≥ ordλ(u) − ǫ.

13 Proof : We write u as in 23, and observe that the terms on the right hand have pairwise distinct orders at λ,

k q ord (z − λ) (uk) = k + q · ord(uk), (27) since 0 ≤ k < q. Denote byµ the remainder of dividing ord(u) by q. It follows that k q µ q the term in {(z − λ) uk : 0 ≤ k < q} with the lowest order is (z − λ) uµ. Therefore, k q if we rewrite the terms {Dǫ (z − λ) uk : 0 ≤ k < q} on the right hand of 24 as { k (z − λ)k−ǫuq : ǫ ≤ k < q} (due to 25) we again notice that all non-vanishing terms ǫ k have diﬀerent orders again, equal to the order of the corresponding term of u minus ǫ. µ q If the term Dǫ ((z − λ) ) uµ does not vanish, it will have the smallest order among all terms, and in this case

µ q µ q ord(Dǫ(u)) = ord Dǫ ((z − λ) ) uµ = ord (z − λ) uµ − ǫ = ord(u) − ǫ (28)

If, however, that term vanishes, some other term with greater order (as a consequence of the analysis above) will give ord (Dǫ(u)). In this case, ord (Dǫ(u)) > ord(u) − ǫ and our proof is complete. ✸

3.4 The orders of inverse images and Proof of Theorem 1.3

Let y ∈ F (z) and let x =(x1, x2,...,xn) be a solution of

n n n s−1 ps pj y = f(x)= fi(xi)= bixi + ci,jxi , (29) i=1 i=1 i=1 j=0 X X X X s where xi ∈ F (z), n ≤ q = p , s ∈ N, and bi,ci,j belong to F [z]. We assume that the polynomials {bi : i =1,...,q} are linearly independent over Vs(F ). We will prove the following.

Proposition 3.4 Let F and f(X¯) as above and let X¯ = (x1, x2,...,xn) a solution of y1 f(X¯) = y, with xi ∈ F (z) for all 1 ≤ i ≤ q. Let y = , where y ,y ∈ F [z]. Then, y2 1 2 there is a constant C, depending only on deg(y2) and the coeﬃcients of f, such that, for any pole λ ∈ F of some xk, for all 1 ≤ i ≤ n, we have ordλ(xi) ≥ C.

Proof : Consider 29 and move the terms of lower degree on the right hand side:

n s − q q ps j b1x1 + ··· + bnxn = u := − ci,jxi + y. (30) i=1 j=1 X X

By Corollary 3.2, since {b1,...,bn} are linearly independent over Vs(F ), there exist s integers 0 = ǫ1 < ǫ2 < ··· < ǫn

14 For every ǫ ∈{ǫi :1 ≤ i ≤ n}, we apply the hyperderivative Dǫ on both sides of 30. We get

n n s−1 n q ps−j xi Dǫ(bi)= Dǫ(u)= γi,jxi + γiDǫ(xi)+ Dǫ(y), (31) i=1 i=1 j=1 i=1 X X X X where γi,j,γi all polynomials of ci,j and their hyperderivatives. Consider the system obtained by writing Equation 31, for ǫ = ǫ1, ǫ2,...,ǫn,

n q xi Dǫ1 (bi)= Dǫ1 (u) i=1 X . . (32) n q xi Dǫn (bi)= Dǫn (u) i=1 X q If we consider the variables xi on the left hand side of the system as the unknowns then the determinant of the system is W := det (Dǫi (bj)) =6 0. Let J be an index such that ord(xJ ) ≤ ord(xi) for every i ∈{1,...,n}. Notice that ord(xJ ) < 0 since we assumed that λ is a pole of some xk. Applying Crammer’s rule q for the J-th index, we get that W xJ = Λ, where

b1 ··· bJ−1 u bJ+1 ··· bn D (b ) ··· D (b ) D (u) D (b ) ··· D (b ) ǫ2 1 ǫ2 J−1 ǫ2 ǫ2 J+1 ǫ2 n D (b ) ··· D (b ) D (u) D (b ) ··· D (b ) Λ := ǫ3 1 ǫ3 J−1 ǫ3 ǫ3 J+1 ǫ3 n ......

Dǫn (b1) ··· Dǫn (bJ−1) Dǫn (u) Dǫn (bJ+1) ··· Dǫn (bn)

n s−1 n n n ps−j = δi,jxi + δiDǫk (xi)+ ζkDǫk (y), (33) i=1 j=1 i=1 k k X X X X=1 X=1 where δi,j, δi and ζk are polynomials that can be determined from f. We are now interested in determining bounds for ord of each term on the right hand of 33. ps−j Regarding the terms of the form δi,jxi , where 1 ≤ i ≤ n and 1 ≤ j ≤ s − 1, it holds that − ps j s−1 ord δi,jxi ≥ p ord(xJ ) + ∆, (34) where ∆ := mini,j{ord(δi,j)}, since ord(xi) ≥ ord(xJ ) and ord(xJ ) < 0.

Regarding the terms of the form δiDǫk (xi), where 1 ≤ i ≤ n and 1 ≤ k ≤ n, it holds that s−1 ord(δiDǫk (xi)) ≥ p ord(xJ )+ E − (q − 1), (35)

15 where E := mini{ord(δi)}. To prove this, we have used Theorem 3.3 along with the relations ǫk ≤ q − 1, ord(xi) ≥ ord(xJ ) and ord(xJ ) < 0. ps−j Since ∆, E ≥ 0, all terms δi,j(xi) and δiDǫk (xi) have ordλ greater than or equal to s−1 n p ordλ(xJ )+1−q. Let Φ2 be the order at λ of the term k=1 ζiDǫk (y) appearing in 33. Then, clearly, it holds that Φ2 ≥ − deg(y2) − (q − 1), due to Theorem 3.3. Also observe q P that, from the relation W xJ = Λ, it follows that ordλ(W )+ qordλ(xJ ) = ordλ(Λ). s−1 We now compare Φ2 and p ordλ(xJ )+1 − q and distinguish two cases.

s−1 s−1 • If p ordλ(xJ )+1 − q < Φ2, then ordλ(Λ) ≥ p ordλ(xJ )+1 − q. In this case, we conclude that 1 − q − ord (W ) 1 − q − deg (W ) ord (x ) ≥ λ ≥ z . (36) λ J q − ps−1 q − ps−1

s−1 • If p ordλ(xJ )+1 − q ≥ Φ2, then ordλ(Λ) ≥ Φ2 ≥ − deg(y2) − (q − 1). It follows that 1 − q − deg (W ) − deg(y ) ord (x ) ≥ z 2 . (37) λ J q

Take C to be the minimum of the bounds of ordλ(xJ ) in 36 and 37. Note that ordλ(xi) ≥ ordλ(xJ ) ≥ C for every 1 ≤ i ≤ n and the proof is complete. ✸ Proof :[Proof of Theorem 1.3] It suffices to establish the conclusion of Proposition 3.4 not only for affine poles but also for the pole at infinity. Here is how: Let η ∈ F such that none of the coefficients of f has a zero at η. Apply the automorphism deg(bi) z 7→ 1/(z − η) and observe that coefficients bi map to ˜bi/(z − η) , where ˜bi ∈ F [z]. M Let M := maxi{deg(bi)}. Clear the denominators by multiplying with (z − η) and apply Proposition 3.4 for λ = η. ✸

4 The Proof of Theorem 1.1

We will now prove a series of propositions that will be used to prove Theorem 1.1.

Proposition 4.1 Let f and h be additive polynomials of degree ps, in m and n variables respectively, where m+n = ps. Let H and G be additive polynomials in only F -variables. Assume that, for some N ∈ N and for some e ∈ Fp[z], it holds that 1 R = Im(f)+ Im(h)+ Im ( G). (38) F eN

Then, the formula u ∈ Im(f)+ ImF (H) is equivalent, in R, to the formula φ1(u), deﬁned as 1 ∀x =(x ,...,x ) ∀y∀γ [(γ ∈ F ∧ u = f(x)+ h(y)+ G(γ)) → π (y,γ)], (39) 1 m eN 1 16 where π1(y,γ) is the formula 1 π (y,γ): ∃w∃α[α ∈ F ∧ f(w)+ h(y)+ Gˆ(γ)= H(α)] . (40) 1 eN In the writing of the formulas above we mean that the variables of the tuples x,y,w,α,β and γ are pairwise distinct.

Note that essentially e = s1s2 ··· sν. The proof is similar to the one of Claim 4.2 (iii) of [PZ04]. For completeness, we include a detailed proof in Section A.2. We will be referring to an existential formula φ(u, {vj}j∈J ), as in 5.

Proposition 4.2 With notation as above, the formula φ(u, {vj}j∈J ) is equi-valent, in R, to the formula φ2, given by

φ2 : u ∈ Im(f)+ ImF (H)∧ (41)

∀w ∀β [(β ∈ F ∧ f(w)+ H(β)= u) → π2({vj}j∈J ,w,β)],

where

π2({vj}j∈J ,w,β): ∃t∃γ [γ ∈ F ∧ f(t)+ H(γ)=0 ∧ (42)

{∧j∈J ej(t)+ Gj(γ) =6 vj − ej(w) − Gj(β)} ∧ Pσ(β + γ)].

Here, Pσ(β + γ) denotes the result of substitution in Pσ(α) of the tuple of variables α by the array of variables β + γ where + implies component-wise addition. The variables of the tuples α, β and γ are pairwise distinct. The proof is similar to the one of Claim 4.1 of [PZ04]. For completeness, we include a detailed proof in Section A.3. Note that Proposition 4.2 holds even if the ring R is replaced by any subring of F (z) that contains F [z].

e Proposition 4.3 A bounded existential Lp(z) -formula is equivalent in R to a univer- e sal Lp(z) -formula. Proof : We will prove the proposition for any formula of the form

π({vj}j∈J , β): ∃γ [γ ∈ F ∧ {∧j∈J Gj(β,γ) =6 vj} ∧ Pσ(β,γ)], (43) where vj are terms of Lp(z) and the Gj are additive polynomials in the variables of the tuples of F -variables β and γ. Let M be the maximum of the degrees of the Gj with respect to the variable z. Pick an element Q of Fp[z], which is not divisible by any polynomial that is irreducible in F [z] and invertible in R. For each of the terms vj, we construct the term M i tj := µi,jQ , (44) i=0 X 17 d k where each µi,j is a term of the form µi,j = k=0 µi,j,kz with d := deg(Q), and µi,j,k are new F -variables. For each j ∈ J, let yj be a new variable (ranging over R). P Observe that, by Fact 1.6, for each j ∈ J and any value of the term vj over R, there N+1 are values of the variables of µi,j over F so that vj = tj + Q yj holds true. And, for those values and any value of the variables of the tuples β and γ, the sub-formula Gj(β,γ) =6 vj is equivalent to yj =06 ∨ Gj =6 tj. Hence π is equivalent to the formula

N+1 χ({vj}j∈J , β): ∀µ∀y (β ∈ F ∧ µ ∈ F ∧ {∧j∈J vj = tj + Q yj}) → (45)

(∨K⊂J ({∧j∈K yj =06 }∧{∧j∈J\K yj =0} ∧ πK[vj/tj])), where

• µ stands for the tuple of all variables µi,j,k, y stands for the tuple of variables yj,

• the index K of ∨K ranges over all subsets of the set J of indices in π and,

• for each subset K of J, the formula πK [vj/tj] is the formula that results from π by deleting the inequalities Gj(β,γ) =6 vj for which j ∈ K and replacing each term vj, for which j ∈ J \ K, by tj. We prove the equivalence of π and χ: Assume that π({vj}j∈J , β) is true for some set of valuesv ˜j of the terms vj and β˜ of N+1 the F -variables β. Write eachv ˜j asv ˜j = t˜j + Q y˜j, using Fact 1.6. Let K be the set of indices j for whichy ˜j =6 0. Then, for each j ∈ J \ K, we havey ˜j = 0. Then, since π({v˜j}j, β˜) is true, it holds that there areγ ˜ over F so that the inequalities Gj(β,˜ γ˜) =6 v ˜j hold for j ∈ J \ K, hence, for those j, we havey ˜j = 0 and Gj(β,˜ γ˜) =6 t˜j holds true and Pσ(β,γ) is true; Hence πK [vj/tj] is true. So χ({v˜j}j, β˜) is true. We leave the converse to the reader. Now observe that each sub-formula πK [vj/tj] of π is equivalent in R to a formula of the form Pτ , for some formula τ of Lp. The same holds true for equations of the form H(β) = 0 that may appear in the quantifier free part of π. So the result holds in the desired generality. ✸ Proof :[of Theorem 1.1] By Lemma 2.1, given an additive polynomial f as in Proposition 4.1 or 4.2, there is a proper transformation ξ(Y, δ) such that, setting f˜(y)+G(δ)= f(ξ(Y, δ)), the additive polynomial f˜ is strongly normalized. Hence, since a proper transformation is onto, we may assume that the f appearing in Propositions 4.1 or 4.2 are strongly normalized. Lemma 2.4 implies that, since the additive polynomial f is strongly normalized, there are h and G so that the assumption of Proposition 4.1 is satisfied for e ∈ Fp[z] and N ∈ N. Observe that, due to Theorem 1.3, the variables of w in π1 can be substituted by a finite tuple of F -variables. Hence, formula π1(y,γ) is equivalent to a formula of the form Pτ , for some formula τ of Lp. It follows that φ1(u) is equivalent to e a universal Lp(z) -formula. In Proposition 4.2 observe that, from Theorem 1.3 and the equation f(t)+H(γ)=0 in π2, the variable t may be substituted by a finite tuple of F -variables. Hence the formula π2({vj}j∈J ,w,β) is equivalent in R to a bounded existential formula.

18 e So far we have proved that any existential Lp(z) -formula is equivalent in R to a disjunction of bounded existential formulas, in which equations do not occur. From Proposition 4.3 it follows that formulas π1 of Proposition 4.1 and π2 of Proposition e 4.2 are equivalent to universal Lp(z) -formulas, and actually to ones with no more free variables than those present in π1 and π2 respectively. e By induction on the number of alterations of quantifiers of an arbitrary Lp(z) - formula in prenex form Theorem 1.1 is proved. e For Item 2 of Theorem 1.1, it follows from Item 1 that any Lp(z) -sentence is equivalent to an existential formula like φ in 5, which is, in addition, a sentence. This means that, in this case, the terms u and vj are concrete elements of Fp[z]. By Lemma 2.1 we may assume, without loss of generality, that the additive polynomial f is strongly normalized. Re-enumerate the variables of x so that x =(x1,...,xk, xk+1,...,xm) and xk+1,...,xm are exactly the variables of x which occur in f with non-zero highest degree coefficient. Then, by Theorem 1.3, for any valuex ˜ of the tuple x which is a solution of the equation f + H = u, the heights ofx ˜k+1,..., x˜m are effectively bounded, hence, the variables xk+1,...,xm may be substituted by (existentially quantified) F -variables. Therefore, we may assume that the sentence φ has no equations and amounts to the solvability of the system of inequalities gj +Gj =6 vj, together with Pσ. Clearly, because R is an infinite domain, all inequalities in which some of the variables x1,...,xk occurs with a non-zero coefficient may be satisfied simultaneously. All the inequalities in which none of the variables x1,...,xk occurs is clearly equivalent to a formula of the form Pτ . Hence φ is equivalent in R to a formula of the form Pτ , for some sentence τ of Lp. ✸

Acknowledgments

This research work was supported from Greek and European Union resources, through the National Strategic Reference Framework (NSRF 2014-2020), under the call “Sup- port for researchers with emphasis on young researchers (EDBM103)” and the funded project “Problems of Diophantine Nature in Logic and Number Theory” with code MIS 5048407.

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21 A Appendix - Omitted Proofs

A.1 Proof of Fact 1.6

a Proof : To prove Item 1, consider a u = b ∈ R with a and b coprime polynomials of F [z] and let c ∈ F [z] \{0}. Then b is invertible in R. By the elementary algebra of polynomials we know that there are v′,r′ ∈ F [z], with deg(v′) < deg(b) and deg(r′) < ′ ′ a ′ a ′ ′ deg(c), such that v c + r b = 1. So b = v b c + ar . Divide the polynomial ar by c - using euclidean division of polynomials - to obtain the result. For the uniqueness statement, assume in the above that c is not divisible by any irreducible factor in F [z] which is invertible in R. Assume that there are v,v ˆ in R and r andr ˆ in F [z] such that u = vc + r =vc ˆ +r ˆ. Then 0 = (v − vˆ)c +(r − rˆ) with deg(r − rˆ) < deg(c). Then there is a ˆb ∈ F [z], which is invertible in R, such that ˆb(v − vˆ) ∈ F [z]. Hence 0 = ˆb(v − vˆ)c + ˆb(r − rˆ). Hence c divides ˆb(r − rˆ) in F [z]. But by hypothesis c is co-prime to ˆb. Hence c divides r − rˆ in F [z], so, necessarily r − rˆ = 0. For Item 2, iterate the conclusion of Item 1 N times, each time applying it to the ‘quotient’ v of the previous step. Item 3 follows from Item 1. ✸

A.2 Proof of Proposition 4.1 Proof : (→) Say that for someu ˜ ∈ R, for some valuex ˜ of the tuple of variables x and for some valueα ˜ (over F ) of the tuple of variables α we haveu ˜ = f(˜x)+ H(α). Consider any valuex ˜ of the tuple x, any valuey ˜ of the tuple y and any value (over F ) of the tuple of variablesγ ˜ of the tuple γ for which 1 u˜ = f(˜x) + h(˜y) + eN G(˜γ). Setw ˜ =x ˜ − x˜ where − denotes component-wise subtraction. Then, by the additivity of f we have

1 1 f(˜w)+ h(˜y)+ G(˜γ)= f(˜x − x˜)+ h(˜y)+ G(˜γ)= (46) eN eN 1 f(˜x) − f(˜x)+ h(˜y)+ G(˜γ)=˜u − f(˜x)= H(˜α) . eN

1 (←) By hypothesis there arex ˜,y ˜ andγ ˜ so that u = f(˜x)+h(˜y)+ eN G(˜γ). Letu ˜ ∈ R 1 and assume that φ3(˜u). Then there is aw ˜ and aα ˜ such that f(˜w)+ h(˜y)+ eN G(˜γ)= H(˜α). Setx ˜ =˜x − w˜. By the additivity of f and we have

f(˜x)+ H(˜α)= f(˜x) − f(˜w)+ H(˜α)= (47) 1 1 [˜u − h(˜y) − G(˜γ)] − [H(˜α) − h(˜y) − G(˜γ)] + H(˜α)=˜u , eN eN

henceu ˜ ∈ Im(f)+ ImF (H). ✸

22 A.3 Proof of Proposition 4.2

Proof : Assume thatu ˜ andv ˜j are given values of the terms u and vj, respectively. Assume that for some valuex ˜ of the array of variables x over R and for some valueα ˜ of the array of variables α over F the statement ψ(˜x, α˜) is true in R, with ψ as in 6, i.e.,

f(˜x)+ H(˜α)=˜u ∧j∈J ej(˜x)+ Gj(˜α) =6 v ˜j ∧ Pσ(˜α) (48) holds. Letw ˜ be a tuple of elements of R and β˜ be a tuple of elements of F such that f(˜w)+ H(β˜)=˜u is true in R. Deﬁne t˜ =x ˜ − w˜ andγ ˜ =α ˜ − β˜, where − denotes component-wise subtraction. Then, by the additivity of f, H, ej and Gj we have

f(t˜)+ H(˜γ)= f(˜x) − f(˜w)+ H(˜α) − H(β˜)=˜u − u˜ =0, (49) and for each j ∈ J the following holds:

ej(t˜)+ Gj(˜γ)= ej(˜x)+ Gj(˜α) − ej(˜w) − Gj(β˜) =6 v ˜j − ej(˜w) − Gj(β˜) . (50)

Moreover Pσ(˜α) holds true, hence Pσ(β˜+˜γ) holds true. It follows that R|= φ → φ2. Now assume that φ2 is true in R for the given valuesu ˜ andv ˜j of u and vj, respectively. Sinceu ˜ ∈ Im(f) + ImF (H) there are valuesw ˜ over R and β˜ over F of the variables w and β such that β˜ ∈ F ∧ f(w)+ H(β˜)=u ˜. Since φ2 is true, there are values t˜ over R andγ ˜ over F of the variables t and γ, respectively, so that f(t˜)+ H(˜γ)=0 ∧j∈J ej(t˜)+ Gj(˜γ) =6 vj − ej(˜w) − Gj(β˜) ∧ Pσ(β˜ +˜γ). Deﬁnex ˜ =w ˜ + t˜ andα ˜ = β˜ +˜γ. Obviously ψ(x, α) of 6 is true for the valuesx ˜ over R of x andα ˜ over F of α. It follows that R|= φ2 → φ. ✸

A.4 Outline of the proof of Lemma 2.1 Proof : We present the proof of the statement of the Lemma up to the point that the ˜ m0 resulting f is p-free. Consider a given additive polynomial f(x)= i=1 fi(xi) of degree s si p where each fi(xi) is an additive polynomial of the variable xi only and of degree p jpsi P and coeﬃcients bi ∈ Fp[z]. Assume that the set B = {biz | i = 1,...,m0, 0 ≤ j < s−si p } is linearly dependent in Vs(Fp). Then there are ci,j ∈ Fp[z], not all equal to 0, so s p jpsi that i,j ci,jbiz = 0, where the index j ranges as in the set B. Re-enumerating the indices i we may assume that some c1,j is not equal to 0 and for each i for which there P is a j so that ci,j =6 0 we have s1 ≥ si. Let

− ps s1 −1 − ps s1 µ c := c1,µ z . (51) µ=0 X 23 − ps s1 −1 F Notice that, since the set {1,...,z } is linearly independent in Vs−s1 ( p) (see Fact 1.8) we have c =6 0. We apply the proper transformation: x1 = cy1 + H and, for i =6 1,

− ps si −1 − − ps si j ps1 si xi = yi + ci,j z y1 , (52) j=0 X ℓ−1 where H = α0 + α1z + ··· + αℓ−1z , ℓ is the degree of c and the αk are new and pairwise distinct F -variables. The polynomial that results from the transformation has the form fˆ+ G, where fˆ has variables the unrestricted (i.e. not F -variables) yi and G ps1 is an additive polynomial in the F -variables αk. We observe that the coeﬃcient of y1 s−s s ˆ ps1 p i −1 p jpsi ˆ in f is b1c + i6=1 j=0 ci,jz = 0. Then f has degree in y1 less than the degree of f in x1 and for all i> 1 it has degree in yi equal to the degree of f in xi. Work by induction on theP sumP of the degrees of f in each of its (unrestricted) variables. The proper transformation ξ is onto, by Facts 1.6 and 1.8, so Im(f)= Im(fˆ)+ ImF (G). Now consider a given additive polynomial f, as above, which is p-free. It is easy to see that substituting each variable xi by

− − − − ps si k ps si ps si −1 ps si − xi,0 + ··· + z xi,k + z xi,ps si −1 , (53) where the xi,k are new variables, results in a normalized additive polynomial f˜: The degree of f˜ with respect to each of its variables is s and the set of its leading coeﬃcients is linearly independent in Vs(Fp). By Fact 1.8, f and f˜ have the same image. To convert a normalized additive polynomial to a strongly normalized one takes the application of a sequence of proper transformations - it is left to the reader or see [PZ04], Lemma 3.3. ✸