A factorization theorem for exponential

P. D’Aquino and G. Terzo Seconda Universit`adegli Studi di Napoli

GOAL: We give a factorization theorem for the of exponential polynomials in many variables over an algebraically closed field of characteristic 0 with an expo- nentiation.

1 E-rings

Definition 1. An exponential ring, or E-ring, is a pair (R,E) with R a ring (com- mutative with 1) and E :(R, +) → (U(R), ·) a map of the additive group of R into the multiplicative group of units of R, satisfying 1. E(x + y) = E(x) · E(y) for all x, y ∈ R

2. E(0) = 1.

Example

1. (R, ax), with a > 0, and (C, ex). 2. (R,E) where R is any ring and E(x) = 1 for all x ∈ R.

3. (S[t],E), where S is an E-field of characteristic 0 and S[t] is the ring of formal

power series in t over S. Let f ∈ S[t], where f = r + f1, and r ∈ S, ∞ X n E(f) = E(r) · (f1) /n! n=0 4. K[X]E ring of exponential polynomials over (K,E), an E-ring. (K,E) is an E-field if K is a field.

1 2 E- ring

Let (K,E) be an E-field, the ring of E-polynomials in the indeterminates x = x1, . . . , xn is an E-ring constructed as follows by recursion.

• (Rk, +, ·)k≥−1 are rings;

• (Bk, +)k≥0 are torsion free abelian groups, and for the algebraically closed fields, are also divisible groups;

• (Ek)k≥−1 are partial E-morphisms.

Step 0: Let R−1 = K; R0 = (K[x], +, ·); B0 =< x >, the ideal generated by x. So R0 = R−1 ⊕ B0; E−1 : R−1 −→ R0, is the composition of the initial E-morphism over K with the immersion of K into K[x].

Inductive step: Suppose k ≥ 0 and Rk−1, Rk, Bk and Ek−1 have been defined in such a way that:

Rk = Rk−1 ⊕ Bk,

Ek−1 :(Rk−1, +) → (U(Rk), ·)

U(Rk) = invertible elements of Rk. Let

Bk t :(Bk, +) → (t , ·) be an isomorphism. Define

Bk Rk+1 = Rk[t ] (group ring construction).

• Rk is a subring of Rk+1

b • 1. Bk+1 is the Rk-submodule of Rk+1 freely generated by t where b ∈ Bk,

2. Rk+1 = Rk ⊕ Bk+1 as additive groups

Bk⊕Bk−1 Bk⊕Bk−1⊕Bk−2 Bk⊕...⊕B0 • Rk+1 = Rk−1[t ] = Rk−2[t ] = ...... = K[x][t ]

b • Ek :(Rk, +) → (U(Rk+1), ·) defined as follows Ek(x) = Ek−1(r) · t

2 Get a chain of partial E-rings (domain of Ek+1 = Rk)

R0 ⊂ R1 ⊂ R2 · · · ⊂ Rk ⊂ · · ·

The ring of exponential polynomials is

∞ E [ B0⊕...⊕Bk⊕... K[x] = lim Rk = Rk = K[x][t ] = U[G] k k=0 group ring where

• U = K[x] a UFD

• G = tB0⊕...⊕Bn⊕... a divisible torsion free (G is orderable and a Q-vector space)

Exponentiation on K[x]E is defined as follows

E(f) = Ek(f), if f ∈ Rk and k ∈ N.

Proposition 1. If K is an exponential field of characteristic 0 then the E-polynomial ring K[x]E is an integral domain whose units are of the form E(α), where α ∈ K[x]E.

Definition 2. An exponential polynomial f ∈ K[x]E is irreducible if there are no non-units g and h with f = gh.

3 Associate polynomial

To any exponential polynomial we will associate a classical polynomial.

E Let f ∈ K[x] , f ∈ Rk for some k ∈ N, then

N X αh f = aht , h=1 where ah ∈ K[x] and αh ∈ B0 ⊕ ... ⊕ Bk−1.

Definition 3. The support of f is the Q-vector space generated by α1, . . . , αN , and is denoted by supp(f).

3 • Let {ν1, . . . , νp} be a Q-base for supp(f). Then

p X αi = rijνj j=1 for all i = 1,...,N, and ri,j ∈ Q.

• Wlog we can assume rij ∈ Z. Moreover, we can assume all rij’s positive integers since we can multiply f by an invertible element, i.e. a purely exponential term. Then f is a polynomial in tν1 , . . . , tνp , with coefficients in a UFD, U = K[x]. Let

ν1 νp y1 = t , . . . , yp = t .

If we substitute each αi by its expression in terms of the bases ν1, . . . , νp we transform f into a classical polynomial in the variables y1, . . . , yp

E f ∈ K[x] Q(y1, . . . , yp) ∈ U[y1, . . . , yp]

where U = K[x]. We will refer to Q(y1, . . . , yp) as the associate polynomial of f.

Definition 4. An exponential polynomial f is simple if dim(supp(f)) = 1.

m1 mp In what follows a monomial in y1, . . . , yp is a term y1 · ... · yp with mi ∈ Z.

Definition 5. A classical polynomial Q(y1, . . . , yp) is essentially 1-variable if there are monomials τ1, τ2 in y1, . . . , yp such that Q(y1, . . . , yp) = τ1P (τ2), where P is a polynomial in just one variable.

Example: The polynomial

2 3 9 2 6 Q(x, y) = x y(3x y − 2x y + 1) = τ1P (τ2),

2 3 3 2 where τ1 = x y, τ2 = xy and P (z) = 3z − 2z + 1, is an essentially 1-variable.

4 Remarks:

1. The correspondence between f and Q holds modulo a monomial in y1, . . . , yp which corresponds to an invertible element of K[x]E;

2. f is a simple polynomial iff Q is essentially 1-variable polynomial.

Let Q(y1, . . . , yp) ∈ K[y1, . . . , yp] be an irreducible polynomial over a UFD U. It µ1 µp can happen that for some µ1, . . . , µp ∈ N+,Q(y1 , . . . , yp ) becomes reducible. For example,

Q(x, y) = x − y is irreducible, but Q(x3, y6) = (x − y2)(x2 + xy2 + y4).

Definition 6. A polynomial Q(y1, . . . , yp) is power irreducible over U if for each µ1 µp µ1, . . . , µp ∈ N+,Q(y1 , . . . , yp ) is irreducible.

Definition 7. A polynomial Q(y1, . . . , yp) is primary in yi if the g.c.d. of the expo- nents of yi in all terms of Q is 1.

Definition 8. A polynomial Q(y1, . . . , yp) is primary if it is primary in each vari- able.

Example: • Q(x, y) = 3x2y − 5y3 + x3 is primary in both x and y.

• R(x, y) = 3x2y − 5y3 + x4 is not primary in x since R(x, y) = P (x2, y), where P (x, y) = 3xy − 5y3 + x2

Remark:

If Q(y1, . . . , yp) is a non primary polynomial then there exists a unique p-upla t1, . . . , tp of positive integers such that

t1 tp Q(y1, . . . , yp) = P (y1 , . . . , yp ) where P (y1, . . . , yp) is primary.

5 4 Factorization Theorem

Known results: • Ritt in [8] was the first to consider a factorization theory for exponential polynomials of the following form

α1z αnz f(z) = a1e + ... + ane ,

where ai, αi ∈ C. • Gourin and Macoll in [6], [7] gave refinements of Ritt’s factorization theorem for exponential polynomials of the form

α1z αnz f(z) = p1(z)e + ... + pn(z)e ,

where αi are complex numbers and pi(z) ∈ C[z]. • Only in the mid ’90s van der Poorten and Everest obtained a factorization theorem in a more general context for exponential polynomials of the second form over an algebraically closed field of characteristic 0 with exponentiation.

We generalize the results obtained in [8], [3], [4] to any exponential polynomial with coefficients in an algebraically closed field K of characteristic 0 with exponentiation.

Major issues for exponential polynomial: If fractional powers are permitted:

1. A binomial as y − 1 defined over an algebraically closed field K may have 1 infinitely many factors. Indeed, y k − , where  is a kth root of unity, is a factor for any positive integer k.

2. The polynomial (x − y) becomes reducible

1 1 2 1 1 2 (x − y) = (x 3 − y 3 )(x 3 + x 3 y 3 + y 3 ).

6 There is a correspondence between a factorization of f and a factorization of Q in fractional powers of the variables.

(⇒) It follows from:

Lemma 1. Let f(x) ∈ K[x]E and suppose that f(x) = g(x)·h(x), where g(x), h(x) ∈ K[x]E. Then supp(g), supp(h) are contained in supp(f).

For the proof we use the construction of the E-polynomial ring and the fact that G is an ordered group.

Corollary 1. Suppose f factorizes as f = gh. Let Q, P and R be the associate polynomials of f, g and h, respectively. There is a monomial τ such that Q = P 0R0 where P 0 = P τ, R0 = τ −1R.

(⇐) It follows from:

Theorem 1. Let Q(y1, . . . , yp) be a (primary) irreducible polynomial over U, a unique factorization domain containing all roots of unity. Assume also that Q is not essentially a 1-variable polynomial. Then Q(y1, . . . , yp) has a factorization into primary irreducible polynomials in the ring generated over U by all fractional powers of y1, . . . , yp, and the number of these irreducible factors is finite.

(t) t1 tp Proof: First of all notice that any factorization of Q = Q(y1 , . . . , yp ) for t1, . . . , tp ∈ N gives a factorization in fractional powers of the variables of Q(y1, . . . , yp). t1 tp Hence we will study the factorizations of Q(y1 , . . . , yp ). Claim. There are only finitely many sets of positive integers

t11, . . . , t1p; t21, . . . , t2p; ...... ; tn1, . . . , tnp

ti1 tip such that Q(y1 , . . . , yp ), for i = 1, . . . , n are reducible.

Step 1. For i = 1, . . . , p let i be a primitive tith root of unity. The transformations

k τi : yi 7→ i yi

t1 tp for 0 ≤ k < ti leave Q(y1 , . . . , yp ) unchanged.

t1 tp The next lemma shows that from one irreducible factor of Q(y1 , . . . , yp ) all the others can be obtained via the group G of transformations generated by τi .

7 Lemma 2. Let Q(y1, . . . , yp) be irreducible over U and suppose that for some pos- (t) t1 tp itive integers t1, . . . , tp, Q = Q(y1 , . . . , yp ) is reducible, and Q1,...,Qs are the irreducible factors. Any fixed factor Qi is transformed in all the other factors Qj’s by the above transformations.

(t) Corollary 2. 1. Each Qi in the identity Q = Q1 ...Qs contains all the variables. 2. If Qi is primary then all Qj’s are primary. 3. If Qi is a polynomial in more than two terms then all Qj’s are polynomials in more than two terms.

(t) Step 2. If the irreducible factor Q1 of Q is primary then Q1 also consists of more than two terms.

(t) t1 tp Step 3. If an irreducible factor Q1 of Q = Q(y1 , . . . , yp ) is primary then each tj 2 satisfies the relation tj ≤ M , where M = max(dy1 , . . . , dyp ).

Factorization Theorem for exponential polynomials: An element f 6= 0 of K[x]E, where K is an algebraically closed field of characteristic 0, factors uniquely up to units and associates, as a finite product of irreducibles of K[x], a finite prod- uct of irreducible of K[x]E whose support is of dimension bigger than 1, and a finite E product of elements gj of K[x] , where supp(gj1 ) 6= supp(gj2 ), for j1 6= j2 and whose supports are of dimension 1.

E (EXISTENCE) Let f(x) ∈ K[x] , and Q(y1, . . . , yp) the associate polynomial. Con- sider a factorization of the associate polynomial

Q = Q1 · ... · Qr · P1 · ... · Ps.

1. Qi are essentially 1-variable polynomials;

2. Pj are not essentially 1-variable irreducible polynomials.

1. The Qi’s correspond to the simple factors of f, and we will multiply those which have the same support in order to have all the factors of f of dimension 1 of different support.

2. Each Pj contributes only for a finite number of irreducible factors in fractional n1 np powers of yh. If Pj(y1, . . . , yp) = V (y1 , . . . , yp ), for some n1, . . . , np ∈ N and V (y1, . . . , yp) primary then necessarily, V (y1, . . . , yp) is also irreducible otherwise niνi Pj(y1, . . . , yp) would be reducible. From substitution of yi by t we get a factor of f.

8 Let r1, . . . , rp be positive integers such that

r1 rp V (y1 , . . . , yp ) = V1 · ... · Vq where Vj are primary and irreducible, and q is the maximum number of irreducible niνi/ri primary factors (by Theorem 1 such q exists). By replacing each yh by t in Vk get the irreducible factors of length > 2 of the exponential polynomial f.

(UNIQUENESS) It is enough to prove that if g(x) divides h(x) · l(x) in K[x]E and g(x) has no factor in common with h(x) then g(x) divides l(x). Suppose

g(x) · s(x) = h(x) · l(x) (1) for some s(x) ∈ K[x]E, and (g(x), h(x)) = 1. Let G(y),H(y),L(y),S(y) be the associate polynomials to g(x), h(x), l(x), s(x), respectively. Clearly, (G(y),H(y)) = 1, since any non trivial common factor of G(y) and H(y) would give a non trivial common factor of g(x) and h(x). Equation (1) implies the following relation over a unique factorization domain

G(y) · S(y) = H(y) · L(y). (2)

Since G has no common factors with H then G divides L. This imples that the exponential polynomial g divides l. So the uniqueness of the factorization of f follows.

Consequences in exponential algebra: • If f in K[x]E is irreducible and with support of dimension more than 1 then f is prime. For, if f divides gh then by the factorization theorem f must occur in the factorization of one of g or h.

• The factorization theorem has been used in order to characterize those expo- nential polynomials over exponential fields introduced by Zilber in [9] which have no solutions in the field. This implies Picard’s Little theorem for expo- nential polynomials over a Zilber field (see [1]).

• The factorization of exponential polynomials of the form

α1z αnz f(z) = a1e + ... + ane ,

where ai, αi ∈ C, has been used by van der Poorten and Tijdeman (see [5]) in the analysis of recurrence sequences in connection with Shapiro’s Conjecture.

9 References

[1] P. D’Aquino, A. Macintyre and G. Terzo: Schanuel Nullstellensatz for Zilber fields, Fundamenta Mathematicae 207, (2010), 123-143.

[2] A. Macintyre: Exponential Algebra, in Logic and Algebra. Proceedings of the international conference dedicated to the memory of Roberto Magari, (A. Ursini et al. eds), Lecture Notes in Pure Applied 180, (1991), 191-210.

[3] A. J. van der Poorten: Factorisation in fractional powers, Acta Arithmetica 70, (3), (1995), 287-293.

[4] A. J. van der Poorten and G. R. Everest: Factorisation in the ring of exponen- tial polynomials, Proceedings of the American Mathematical Society, 125, (5), (1997), 1293-1298.

[5] A.J. van der Poorten and R. Tijdeman: On common zeros of exponential poly- nomials L’Enseign. Math. S´erie, 21 (1975) pp. 5767.

[6] E. Gourin: On irreducible polynomials in several variables which become re- ducible when the variables are replaced by powers of themselves, Transactions of the American Mathematical Society 32, (1930), 485-501.

[7] L. A. MacColl: A factorization theory for polynomials in x and in functions eαx, Bulletin American Mathematical Society, (41), (1935), 104-109.

Pn αiz [8] J.F. Ritt: A factorization theorem of functions i=1 aie , Transactions of American Mathematical Society 29, (1927), 584-596.

[9] B. Zilber: Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, 132, (1), (2004), 67-95.

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