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DEFINING MULTIPLICATION FOR POLYNOMIALS OVER A FINITE
FIELD
MICHEL RIGO AND LAURENT WAXWEILER
Abstract. Let Pand Qbe two non-zero multiplicatively independent polynomials with coef-
ficients in a finite field F. Adapting a result of R. Villemaire, we show that multiplication of
polynomials is a ternary relation {(A, B, C)∈F[X]|A.B =C}definable by a first-order for-
mula in a suitable structure containing both functions VPand VQwhere VA(B) is defined as the
greatest power of Adividing B. Such a result has to be considered in the context of a possible
analogue of Cobham’s theorem for sets of polynomials whose P-expansions are recognized by
some finite automaton.
1. Introduction
All along this paper, Fis a finite field and F[X] is the ring of polynomials over F. Let P, Q
be two non-constant polynomials. If the only integers a, b such that Pa=Qbare a=b= 0,
then Pand Qare multiplicatively independent. Let P, Q be two polynomials. We write P≺Q,
if deg P < deg Q(,≻and are defined accordingly). As usual, we set deg 0 = −∞. The
set of powers of Pis PN:= {Pn|n∈N}. Let Pbe a non-constant polynomial. The map
VP:F[X]→PNis defined by VP(0) = 1 and, for all non-zero polynomials A,VP(A) is the largest
power of Pdividing A.
We consider the structure hF[X],+,≺,(·C:C∈F[X]), VPi. In particular, relying on techniques
based on automata, the first-order theory of this structure is known to be decidable [8] and similar
structures have been recently and independently considered by A. Sirokofskich in [10]. One can
notice that we equip the structure with multiplication by a fixed polynomial C, that is, maps of
the kind A7→ C·A.
Interestingly some sets of polynomials can be defined within such a first-order structure. We
can write well-formed formulas where variables are ranging over F[X], using usual logical connec-
tives ∧,∨,¬,→,↔, existential and universal quantifiers (∃Q) and (∀Q) applied only to variables
and also the specific operations of the structure: addition of polynomials, comparison of degree,
multiplication by a fixed polynomial and the map VPdefined above. Let ϕ(R1,...,Rk) be a for-
mula with kfree variables, i.e., not in the scope of any quantifier. This formula defines a subset
of k-tuples of polynomials for which the formula is satisfied:
Mϕ={(Q1,...,Qk)∈(F[X])k| hF[X],+,≺,(·C:C∈F[X]), VPi |=ϕ(Q1,...,Qk)}.
Note that if Pand Qare multiplicatively dependent polynomials of degree at least one, then the
structures hF[X],+,≺,(·C:C∈F[X]), VPiand hF[X],+,≺,(·C:C∈F[X]), VQiare equivalent:
they give exactly the same definable sets. Let us consider a few examples where F=Z/2Z. The
set of polynomials in (Z/2Z)[X] which are divisible by X2is defined by ϕ(A)≡(∃Q)(A=X2·Q).
The set of powers of Xis definable in the structure hF[X],+,≺,(·C:C∈F[X]), VXiby the
formula ϕ(A)≡VX(A) = A.
It turns out that set of polynomials definable in the structure hF[X],+,≺,(·C:C∈F[X]), VPi
are exactly the so-called P-recognizable sets, i.e., sets of polynomials whose P-expansions are finite
words recognized by some finite automaton over a finite alphabet, see again [8]. Indeed, if Pis a
non-constant polynomial, then any non-zero polynomial A∈F[X] can be written in a unique way
as A=Pℓ
i=0 CiPiwith Cℓ6= 0 and polynomials C0,...,Cℓof degree less than the degree of P.
Hence this logical characterization provides us with a useful tool to studying these recognizable
sets of polynomials as initiated in [7].
1

2 MICHEL RIGO AND LAURENT WAXWEILER
The main result of this paper appeared first in the unpublished Ph.D. thesis of the second
author [13] and is the following one.
Theorem 1. Let Pand Qbe two multiplicatively independent polynomials with coefficients in a
finite field F. Then multiplication of polynomials is a ternary relation
{(A, B, C)∈F[X]|A.B =C}
definable by a first-order formula in hF[X],+,≺,(·C:C∈F[X]), VP, VQi.
This means that adding multiplication to the structure hF[X],+,≺,(·C:C∈F[X]), VP, VQi
does not lead to any new first-order definable set of polynomials. It also turns out that any map
VRcan also be defined within this structure.
A result similar to Theorem 1 has been obtained in [12] for the integers and the structure
hN,+, Vk, Vℓiwhere Vk(n) is the largest power of k≥2 dividing n∈N≥1. Our proofs and
guidelines rely on those originally developed by Villemaire. We consider that our work is not a mere
translation of the original ones: non-trivial adaptations, complements and slight corrections had
to be made. Furthermore, the extra arguments that we will present can also be translated to the
original context of integers. Also, such a result can provide some insight about the recognizability
of sets of polynomials over a finite ring which has still to be studied further.
1.1. Motivations. A set Xof integers is said to be p-recognizable, if the set of base pexpansions
of the elements in Xis a regular language over the alphabet of digits {0,...,p −1}(that is
accepted by some finite automaton). A set Xis ultimately periodic if there exist N, q > 1 such
that, for all n≥N,n∈X⇔n+q∈X. Villemaire’s work can be related to the famous
theorem of Cobham [2]: if a set Xof integers is both p-recognizable and q-recognizable with p
and qbeing multiplicatively independent integer bases, then Xis ultimately periodic. Cobham’s
theorem has given a major impulse to studying recognizable sets of integers [1, 3]. In this context,
Villemaire’s work has led to interesting developments concerning p-recognizable sets in a logical
setting [5]. In this context, the work of Semenov and Muchnik have also to be mentioned [9, 6].
Therefore, as for the integer case, we hope that our result could shed some new light on a possible
analogue of Cobham’s theorem in the context of sets of polynomials over a finite field which
are P-recognizable for all polynomial base P. Up to now, p-recognizable or p-automatic sets of
polynomials over a finite field have reveal more properties than those observed for integers and
deserve further investigations [7, 8].
1.2. Organization. This paper is organized as follows. In Section 2, we give a few definitions
needed in the paper. Section 3 is devoted to the following construction. Given two multiplicatively
independent polynomials S, T , we define two multiplicatively independent polynomials P, Q which
are respectively power of Sand T, some increasing map F:PN→PNand a finite partition
(Ei)i∈{1,...,c}of PNsuch that for all i∈ {1,...,c}and all A, B ∈Ei,A≺B→ ∃C∈(PN\
F(PN)) : F(A)≺C≺F(B). Furthermore, we ensure that Fand (Ei)i∈{1,...,c}are definable in
hF[X],+,≺,(·C:C∈F[X]), VP, VQi. In Section 4 we assume the existence of such a map Fand
partition (Ei)i∈{1,...,c}to define multiplication in a enlarged structure where Fand (Ei)i∈{1,...,c}
have been added. Finally, we collect in Section 5 the results of the previous two sections ti get a
proof of the main theorem.
2. A few definitions
We denote respectively by SPand SP,Q the structures hF[X],+,≺,(·C:C∈F[X]), VPiand
hF[X],+,≺,(·C:C∈F[X]), VP, VQi. The terms in SPare variables and constants (when the
structure contains the function VPthen the constant 1 is easily defined by A= 1 ≡VP(0) = A
and the constant 0 by A= 0 ≡(∀B)(A≺B∨A=B)) ranging over F[X] or if Q, R are terms and
Cis a fixed polynomial, then Q+R,Q·C,VP(Q) are terms. If Q, R are terms, then Q=Rand
Q≺Rare atomic formulas. Then one can construct arbitrary formulas using logical connectives
∨,∧,¬,→,↔and quantifiers ∃Qand ∀Qwhere Qis a variable.

DEFINING MULTIPLICATION FOR POLYNOMIALS OVER A FINITE FIELD 3
Definition 1. Let P, Q ∈F[X] be non-constant polynomial. A subset Tof (F[X])dis P-definable
(resp. (P, Q)-definable) if there exists a first-order formula φ(A1,...,Ad) in the language SP(resp.
SP,Q) which is satisfied if and only if (A1,...,Ad) belongs to T. A k-ary relation Tis simply
a subset of (F[X])kand one can define P-definable (resp. (P, Q)-definable) relation accordingly.
A map from (F[X])kto (F[X])ℓis a (k+ℓ)-ary relation. So P-definable (resp. (P, Q)-definable)
maps are defined accordingly.
Definition 2. Let k∈Fbe non-zero. The map lcmk:F[X]×F[X]→F[X] is defined as follows.
If Aor Bis zero, then lcmk(A, B) = 0. Otherwise, lcmk(A, B) is the least common multiple of
the polynomials Aand Bhaving kas leading coefficient.
Definition 3. Any non-zero polynomial A∈F[X] can be written in a unique way as A=
Pℓ
i=0 CiPiwith Cℓ6= 0 and polynomials C0,...,Cℓof degree less than the degree of P. The word
Cℓ···C0over the finite alphabet of polynomials of degree less than deg(P) is the P-expansion or
P-representation of A. It is denoted repP(A). We say that the P-expansion of Ahas coefficient
Cjoccurring for Pj. By convention, the P-expansion of the zero polynomial is the empty word.
Definition 4. The binary relation XP,J (A, B) is true if and only if Ais a power Pand in the
P-representation of B,Aoccurs with Jas coefficient, J≺P. It is P-definable by the formula
(VP(A) = A)∧(∃U)(∃V)(B=U+J·A+V∧V≺A∧(U= 0 ∨A≺VP(U))).
Definition 5. Let k≥1 be an integer. For all A∈F[X], Degk(A) is true if and only if
deg(A)≡0 mod k. Let Cbe a non-zero polynomial. For all A∈F[X], PreC(A) is true if and
only repX(C) is a prefix of repX(A).
Proposition 2. [8] Let Pbe a non-constant polynomial. Let k≥1be an integer. Let Cbe a
non-zero polynomial. The unary relations Degkand PreCare P-definable.
3. A partition of PN
The aim of this section is to prove Theorem 6, namely that if S, T are two multiplicatively
independent polynomials, then there exist Pand Qwhich are respectively power of Sand T(or
Tand S, see (5) for details), some (P, Q)-definable increasing map F:PN→PNand a finite
partition (Ei)i∈{1,...,c}of PNsuch that for all i∈ {1,...,c}and all A, B ∈Ei,
A≺B→ ∃C∈(PN\F(PN)) : F(A)≺C≺F(B).
To that end, we consider decomposition of polynomials into a product of irreducible polynomials
and we mainly have to consider two particular cases depending on the decompositions discussed
in Propositions 3 and 4.
Lemma 1. Let k∈Fbe non-zero. Let P, Q be multiplicatively independent polynomials of the
kind P=P0Rand Q=Q0Rwith P0, Q0, R pairwise coprime polynomials such that P0Q0.
Then the map lcmkrestricted to the domain PN×QNis (P, Q)-definable.
Proof. Consider the graph {(A, B, C)|(A, B)∈PN×QN,lcmk(A, B) = C}of the map lcmk
restricted to PN×QN. This set can be defined by the formula:
VP(A) = VP(C) = A∧VQ(B) = VQ(C) = B
∧(∀T)(VP(T) = A∧VQ(T) = B→TC)∧DebK(C)
where Kis the constant polynomial corresponding to the element k∈F.
Proposition 3. Let P, Q be multiplicatively independent polynomials of the kind P=P0Rand
Q=Q0Rwith P0, Q0, R pairwise coprime polynomials such that P0Q0. Then there exist a
constant c∈N≥1and a (P, Q)-definable map F:PN→PNincreasing with respect to the order
relation ≺and such that, for all k∈N,
Pc+1 ·F(Pk)F(Pk+c).

4 MICHEL RIGO AND LAURENT WAXWEILER
Proof. Let G:PN→QNmapping A∈PNto the smallest power Qkof Qsuch that Qk≻A. The
graph {(A, B)|G(A) = B}of this map is (P, Q)-definable by the formula:
VP(A) = A∧VQ(B) = B∧B≻A∧(∀C)(VQ(C) = C∧C≻A→CB).
In the same way, the function H:F[X]→PNmapping A∈F[X] to the smallest power Pkof P
such that Pk≻Ais also (P, Q)-definable.
Now using the above lemma and composing (P, Q)-definable maps, the map
F:PN→PN, A 7→ H(lcm1(A, G(A)))
is thus (P, Q)-definable.
Let v(k) be the smallest integer such that Pk≺Qv(k). Since P=P0RQ0R=Q, for all
k∈N, we have v(k) + 1 ≤v(k+ 1).
We write A≃B, if deg(A) = deg(B), i.e., AB∧BA. Let us first prove that Fis
increasing, i.e., that, for all k,F(Pk+1)≻F(Pk). We have
F(Pk+1) = H(lcm1(Pk+1, Qv(k+1))) ≃H(Pk+1
0Qv(k+1) 0Rv(k+1))
H(Pk+1
0Qv(k)+1 0Rv(k)+1)H(P P k
0Qv(k)
0Rv(k)).
Since H(P P k
0Qv(k)
0Rv(k)) = P·H(Pk
0Qv(k)
0Rv(k))≻H(Pk
0Qv(k)
0Rv(k)), we get
F(Pk+1)≻H(lcm1(Pk, Qv(k))) = F(Pk).
To conclude with this proof, we have to show that there exists a constant csuch that, for all k∈N,
Pc+1 ·F(Pk)F(Pk+c). Observe that
Pc+1 ·H(lcm1(Pk, G(Pk))) H(lcm1(Pk+c, G(Pk+c)))
holds true whenever Pc+2 ·lcm1(Pk, G(Pk)) lcm1(Pk+c, G(Pk+c)). This latter condition is
equivalent to
(1) P2
0Qv(k)
0Rv(k)+c+2 Qv(k+c) 0Rv(k+c)
which in turn holds whenever P2
0Qv(k)
0Rv(k)+c+2 Qv(k)+c 0Rv(k)+c. We have just shown that (1)
or 2 deg(P0) + 2 deg(R)≤cdeg(Q0) imply Pc+1 ·F(Pk)F(Pk+c).
If deg(Q0)≥1, then the condition is satisfied for any large enough c.
Now consider the case where Q0is a constant in F. In that case, since deg(R) = deg(Q),
condition (1) becomes 2 deg(P0)+(v(k)+c+2) deg(Q)≤v(k+c) deg(Q). We have, by definition of
v,Pk+c≺Qv(k+c)and Qv(k)−1Pkfor all k, c. Therefore, we get (k+c) deg(P)< v(k+c) deg(Q)
and (v(k)−1) deg(Q)≤kdeg(P) for all k, c. From these two inequalities, we conclude that
(2) cdeg(P) = (k+c−k) deg(P)<(v(k+c)−v(k) + 1) deg(Q).
Note that deg(P0)≥1, because otherwise P=kR and Q=ℓR for some non-zero k, ℓ ∈Fand
one can conclude that Pand Qare multiplicatively dependent. Therefore, for large enough c,
(3) c(deg(P)−deg(Q)) ≥2 deg(P0) + 3 deg(Q).
From (2), we obtain that
v(k+c) deg(Q)> c(deg(P)−deg(Q)) + (v(k) + c−1) deg(Q)
and we conclude the proof. Using (3), we get v(k+c) deg(Q)≥2 deg(P0)+(v(k)+c+2) deg(Q). 
Proposition 4. Let P, Q be two polynomials having decomposition into irreducible polynomials of
the kind
P=
m
Y
i=1
Pγi
i
ℓ
Y
i=1
Rαi
iand Q=
n
Y
i=1
Qδi
i
ℓ
Y
i=1
Rβi
i
where the polynomials P1,...,Pm, Q1,...,Qn, R1,...,Rℓare pairwise coprime, m, n, ℓ ∈N,ℓ≥2
and
1 = α1
β1
≤ · · · ≤ αi
βi
≤ · · · ≤ αℓ
βℓ
=θ, with θ > 1.

DEFINING MULTIPLICATION FOR POLYNOMIALS OVER A FINITE FIELD 5
Then the map F:PN→PN, P k7→ P⌈θk⌉is increasing with respect to the order relation ≺and
there exists a constant c∈N≥1such that, for all k∈N,Pc+1 ·F(Pk)F(Pk+c).
Proof. For all c, k ∈N, we have
F(Pk+c) = P⌈θ(k+c)⌉P⌈θk+⌊θc⌋⌉ =F(Pk)·P⌊θc⌋.
Taking c= 1, leads to F(Pk+1)≻F(Pk) showing that Fis increasing. Fix any integer csuch
that c≥1/(θ−1). Hence we get ⌊θc⌋ ≥ ⌊1 + c⌋and F(Pk+c)F(Pk)·P⌊θc⌋F(Pk)·Pc+1.
Proposition 5. Let P, Q be two polynomials satisfying the assumptions of Proposition 4. The
map F:PN→PN, P k7→ P⌈θk⌉defined in Proposition 4 is (P, Q)-definable.
Proof. First the map G:PN→QN, P k7→ Q⌈θk⌉is (P, Q)-definable. We have B=G(A) if and
only if
VP(A) = A∧VQ(B) = B∧(∀T)VP(T
ℓ
Y
i=1
Ri)A∧VQ(T)B→VP(T)A∧
(∀C)VQ(C) = C∧(∀T)VP(T
ℓ
Y
i=1
Ri)A∧VQ(T)C→VP(T)A→CB.
Indeed, A=Pkand B=Qrfor some integers k, r. Any non-zero polynomial Tcan be written
T=T0
m
Y
i=1
Pγ′
i
i
n
Y
i=1
Qδ′
i
i
ℓ
Y
i=1
Rǫi
i
where T0is a polynomial coprime with all Pi, Qi, Riand γ′
i, δ′ i, ǫiare non-negative integers. Note
that
VP(T) = VPm
Y
i=1
(Pγi
i)γ′
i/γi
ℓ
Y
i=1
(Rαi
i)ǫi/αi=Pmin⌊γ′
i/γi⌋,⌊ǫj/αj⌋|1≤i≤m,1≤j≤ℓ,
VQ(T) = Qmin⌊δ′
i/δi⌋,⌊ǫj/βj⌋|1≤i≤n,1≤j≤ℓ,
VP(T
ℓ
Y
i=1
Ri) = Pmin⌊γ′
i/γi⌋,⌊(1+ǫj)/αj⌋|1≤i≤m,1≤j≤ℓ.
The implication VP(TQℓ
i=1 Ri)A∧VQ(T)B→VP(T)Ais equivalent to
min⌊γ′
i/γi⌋,⌊(1 + ǫj)/αj⌋ | 1≤i≤m, 1≤j≤ℓ≥k
∧min⌊δ′
i/δi⌋,⌊ǫj/βj⌋ | 1≤i≤n, 1≤j≤ℓ≥r
→min⌊γ′
i/γi⌋,⌊ǫj/αj⌋ | 1≤i≤m, 1≤j≤ℓ≥k
and it is satisfied, for all possible T0, γ′
1,...,γ′ m, δ′ 1,...,δ′ n, ǫ1,...,ǫℓdescribing the polynomial T,
if and only if the following implication holds for all ǫ1,...,ǫℓ:
min⌊(1 + ǫj)/αj⌋ | 1≤j≤ℓ≥k∧min⌊ǫj/βj⌋ | 1≤j≤ℓ≥r
→min⌊ǫj/αj⌋ | 1≤j≤ℓ≥k.
This is equivalent to the fact that there is no j∈ {1,...,ℓ}and no ǫjsuch that (1 + ǫj)/αj≥
k > ǫj/αjand ǫj/βj≥r, i.e., such that kαj−1 = ǫj≥rβj. This latter inequalities occur for no
j∈ {1,...,ℓ}and no ǫjif and only if r/k ≥max{αj/βj|1≤j≤ℓ}=θ. In other words, the
middle part of the formula describes that Bis a power of Qhaving an exponent at least kθ. In
the same way, the last part of the formula means that any polynomial Cwhich is a power of Q
having an exponent at least kθ is such that CB.