Luis H Gallardo

Olivier Rahavandrainy

ALL EVEN (UNITARY) PERFECT POLYNOMIALS OVER F 2 WITH ONLY MERSENNE PRIMES AS ODD DIVISORS

Keywords: Sum of divisors, polynomials, finite fields, characteristic 2. 2010 Mathematics Subject Classification. Primary: 11T55. Secondary: 11T06. DOI

We address an arithmetic problem in the ring F 2 [x]. We prove that the only (unitary) perfect polynomials over F 2 that are products of x, x + 1 and of Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of M 2h+1 + 1, for a Mersenne prime M and for a positive integer h.

Introduction

Let A ∈ F 2 [x] be a nonzero binary polynomial. Let σ(A) denote the sum of all divisors of A (including 1 and A). If σ(A) = A, then one says that A is a one-ring ( [START_REF] Canaday | The sum of the divisors of a polynomial[END_REF]) or in other words, A is perfect ( [START_REF] Beard | Perfect polynomials over GF (q)[END_REF]). In addition to polynomials of the form (x 2 +x) 2 n -1 , with some positive integer n, E. F. Canaday ([5]) discovered eleven non-splitting perfect polynomials (cf. Notation): T 1 , . . . , T 9 and C 1 , C 2 . The T j 's are divisible only by x, x + 1 and by irreducible polynomials of the form U a,b := x a (x + 1) b + 1, for some positive integers a, b. The last two C 1 and C 2 are divisible by x 4 + x + 1 which is not of the form U a,b . The parallel with the integer case is then natural to be considered. We know that all perfect numbers are of the form 2 m (2 m -1), where m is a prime number and 2 m -1 is a Mersenne prime number. So, we may consider the following notions. We say ( [START_REF] Gallardo | Even perfect polynomials over F 2 with four prime factors[END_REF]) that a binary polynomial is even if it has a linear factor. It is odd, otherwise. We ( [START_REF] Gallardo | On Mersenne polynomials over F 2[END_REF]) also define a Mersenne prime (polynomial) over F 2 as an irreducible polynomial of the above form U a,b . The name comes as an analogue of the integral Mersenne primes, taking x a (x + 1) b as an analogue of the prime power 2 a+b .

Note that the notion of Mersenne prime polynomial is only useful over F 2 , whereas one may consider the "parity" of a polynomial over any finite field.

Unitary perfect polynomials are defined and studied in several directions by J. T. B. Beard Jr. et al. ( [START_REF] Beard | Perfect polynomials revisited[END_REF], [START_REF] Beard | Unitary perfect polynomials over GF (q)[END_REF], [START_REF] Beard | Perfect polynomials over GF (q)[END_REF]). As over the integers, for A ∈ F 2 [x], a divisor D of A is unitary if gcd(D, A/D) = 1. Let σ * (A) denote the sum of all unitary divisors of A (including 1 and A). If σ * (A) = A, then A is unitary perfect.

We say that a (unitary) perfect polynomial is indecomposable if it is not a product of two coprime nonconstant (unitary) perfect polynomials.

Any unitary perfect polynomial is even (Lemma 3.4). The known ones, which are only divisible by Mersenne primes (as odd factors), belong to the equivalence classes (see Lemma 3.5) of B 1 , . . . , B 9 (see Notation). The other ones (which are divisible by non-Mersenne primes) belong to several different (perhaps, infinitely many) classes (see [START_REF] Beard | Unitary perfect polynomials over GF (q)[END_REF] and [START_REF] Rahavandrainy | Familles de polynômes unitairement parfaits sur F 2[END_REF]).

Since a few moments, we would like to continue this investigation (with more or less success). In particular, we want to find all non-splitting (unitary) perfect binary polynomials which are only divisible by x, x + 1 and by Mersenne primes. Some results are obtained ( [START_REF] Gallardo | On even (unitary) perfect polynomials over F 2[END_REF], Theorems 1.1 and 1.3) but they are not complete. The main obstacle is the fact that we cannot understand how M 2h+1 + 1 = (M + 1) σ(M 2h ) factors over F 2 , for a Mersenne prime M and a positive integer h. We have formulated ( [START_REF] Gallardo | On Mersenne polynomials over F 2[END_REF]) a conjecture about that (Conjecture 4.1). The further we make progress on that conjecture, the better we reach our goal. Conjecture 4.1 is already proved under some conditions on M and h ([10, Theorem 1.4]). In this paper, we continue working toward its proof with some new conditions on M and h, where the sets M and ∆ defined below intersect. We get Proposition 1.1 which in turn, allows us to obtain Theorems 1.1 and 1.2. The study of Mersenne primes have some interest. For example, we have established ( [START_REF] Gallardo | On Mersenne polynomials over F 2[END_REF]Theorem 1.3]) that if gcd(a, b) = 1, then U a,b = x a (x + 1) b + 1 has exactly the same number of irreducible divisors as the trinomial x a+b + x b + 1. In particular, they are both irreducible or both not irreducible. So, they would be useful in the domain of error-correcting codes. It is convenient to fix some notation.

Notation

• The set of integers (resp. of nonnegative integers, of positive integers) is denoted by Z (resp. N, N * ).

• For S, T ∈ F 2 [x] and for m ∈ N * , S m | T (resp. S m ∥T ) means that S divides T (resp. S m | T but S m+1 ∤ T ). We also denote by S the polynomial defined as S(x) = S(x + 1) and by val x (S) (resp. val x+1 (S)) the valuation of S, at x (resp. at x + 1).

• We put

M 1 = 1 + x(x + 1), M 2 = 1 + x(x + 1) 2 , M 3 = 1 + x(x + 1) 3 , T 1 = x 2 (x + 1)M 1 , T 3 = x 4 (x + 1) 3 M 3 , T 2 = T 1 , T 4 = T 3 , T 5 = x 4 (x + 1) 4 M 3 M 3 = T 5 , T 6 = x 6 (x + 1) 3 M 2 M 2 , T 7 = T 6 , T 8 = x 4 (x + 1) 6 M 2 M 2 M 3 , T 9 = T 8 , C 1 = x 2 (x + 1)M 2 1 (x 4 + x + 1), C 2 = C 1 . B 1 = x 3 (x + 1) 3 M 2 1 , B 2 = x 3 (x + 1) 2 M 1 , B 3 = x 5 (x + 1) 4 M 3 , B 4 = x 7 (x + 1) 4 M 2 M 2 , B 5 = x 5 (x + 1) 6 M 2 1 M 3 , B 6 = x 5 (x + 1) 5 M 3 M 3 , B 7 = x 7 (x + 1) 7 M 2 2 M 2 2 , B 8 = x 7 (x + 1) 6 M 2 1 M 2 M 2 , B 9 = x 7 (x + 1) 5 M 2 M 2 M 3 .
• The following sets play important roles:

M = {M 1 , M 2 , M 2 , M 3 , M 3 }, P = {T 1 , . . . , T 9 }, P u = {B 1 , . . . , B 9 } ∆ 1 = {p ∈ N * : p is a Mersenne prime}, ∆ 2 = {p ∈ N * :
p is prime and ord p (2) ≡ 0 mod 8}, where ord p (2) denotes the order of 2 in F p \ {0},

∆ = ∆ 1 ∪ ∆ 2 .
In particular, ∆ contains all Fermat primes greater than 5.

Throughout this paper, we always suppose that any (unitary) perfect polynomial is indecomposable. We have often used Maple software for computations. Our main results are the following.

(i) (M ∈ {M 1 , M 3 , M 3 }) or (M ∈ {M 2 , M 2 } and h ≥ 2).
(ii) M ̸ ∈ M and 2h + 1 is divisible by a prime number p lying in ∆ \ {7}.

Theorem 1.1. Let A = x a (x + 1) b i∈I P h i i ∈ F 2 [
x] be such that each P i is a Mersenne prime and a, b, h i ∈ N * . Then A is perfect if and only if A ∈ P.

Theorem 1.2. Let

A = x a (x + 1) b i∈I P h i i ∈ F 2 [
x] be such that each P i is a Mersenne prime and a, b, h i ∈ N * . Then A is unitary perfect if and only if A = B 2 n , for some n ∈ N and B ∈ P u .

We first prove the two theorems before the proposition.

2. Proof of Theorem 1.1

Sufficiencies are obtained by direct computations. For the necessities, we shall apply Lemma 2.3 and Proposition 2.1. We fix:

A = x a (x + 1) b i∈I P h i i = A 1 A 2 ,
where a, b, h i ∈ N, P i is a Mersenne prime,

A 1 = x a (x + 1) b P i ∈M P h i i and A 2 = P j ̸ ∈M P h j j . Lemma 2.1. If A is perfect, then σ(x a ), σ((x + 1) b ) and each σ(P i h i ), with i ∈ I, are only divisible by x, x + 1 or by Mersenne primes. Proof. Since σ is multiplicative, σ(A) = σ(x a )σ((x + 1) b ) i∈I σ(P i h i ). Any divisor of σ(x a ), σ((x + 1) b ) and σ(P i h i ) divides σ(A) = A. □ Lemma 2.2 ([4], Lemma 2).
A polynomial S is perfect if and only if for any irreducible polynomial P and for any m 1 , m 2 ∈ N * , we have:

(P m 1 ∥S, P m 2 ∥σ(S)) ⇒ m 1 = m 2 .
Example 2.1 (useful for Proposition 2.1).

The polynomial

S 1 = x 13 (x + 1) 2 M 3 1 M 2 2 M 2 2 M 3 M 3 is not perfect because x 13 ∥S 1 and
x 7 ∥σ(S 1 ).

Lemma 2.3 ([8], Theorem 1.1). If h i = 2 n i -1 for any i ∈ I, then A ∈ P.
We get from Theorem 8 in [START_REF] Canaday | The sum of the divisors of a polynomial[END_REF] and from Proposition 1.1.

Lemma 2.4.

(i) If h ∈ N * and if σ(x 2h
) is only divisible by Mersenne primes, then 2h ∈ {2, 4, 6} and all its divisors lie in M. More precisely, σ(

x 2 ) = M 1 = σ((x + 1) 2 ), σ(x 4 ) = M 3 , σ((x + 1) 4 ) = M 3 and σ(x 6 ) = M 2 M 2 = σ((x + 1) 6 ). (ii) Let M ∈ M and h ∈ N * be such that σ(M 2h ) is only divisible by Mersenne primes, then 2h = 2, M ∈ {M 2 , M 2 } and σ(M 2 ) ∈ {M 1 M 3 , M 1 M 3 }.
We dress from Lemma 2.4, the following table of all the forms of a, b, P i and h i which satisfy Lemma 2.1, if

P i ∈ M and if h i ̸ = 2 n i -1. Table 2.1. a σ(x a ) 3 • 2 n -1 (x + 1) 2 n -1 M 1 2 n 5 • 2 n -1 (x + 1) 2 n -1 M 3 2 n 7 • 2 n -1 (x + 1) 2 n -1 M 2 2 n M 2 2 n b σ((x + 1) b ) 3 • 2 m -1 x 2 m -1 M 1 2 m 5 • 2 m -1 x 2 m -1 M 3 2 m 7 • 2 m -1 x 2 m -1 M 2 2 m M 2 P i h i σ(P i h i ) M 2 3 • 2 n i -1 (1 + M 2 ) 2 n i -1 M 1 2 n i M 3 2 n i M 2 3 • 2 n i -1 (1 + M 2 ) 2 n i -1 M 1 2 n i M 3 2 n i Corollary 2.1. Suppose that A 1 is perfect. Then, neither M 2 nor M 2 divides σ(P i h i ) if P i ∈ M. Moreover, M 2 divides A 1 whenever M 2 divides A 1 and their exponents (in A 1 ) are equal. Proof. The first statement follows from Lemma 2.4-(ii). Now, if M 2 divides A 1 = σ(A 1 ), then M 2 divides σ(x a ) σ((x + 1) b ) P i ∈M σ(P h i i ). Hence, M 2 divides σ(x a )σ((x + 1) b ). Table 2.1 shows that a or b is of the form 7 • 2 n -1, where n ∈ N. So, M 2 divides σ(A 1 ) = A 1 . It suffices to consider two cases. If a = 7 • 2 n -1 and b = 7 • 2 m -1, then M 2 ℓ ∥A 1 and M 2 ℓ ∥A 1 , with ℓ = 2 n + 2 m . If a = 7 • 2 n -1 and (b = 3 • 2 m -1 or b = 5 • 2 m -1), then M 2 ℓ ∥A 1 and M 2 ℓ ∥A 1 , with ℓ = 2 n . □ Lemma 2.5. If P is a Mersenne prime divisor of σ(A 1 ), then P, P ∈ {M 1 , M 2 , M 3 }.
Proof. One has: σ(A 1 ) = σ(x a )σ((x + 1) b )

P i ∈M σ(P h i i ). If P divides σ(x a )σ((x + 1) b ), then P ∈ M, by Lemma 2.4-(i). If P divides σ(P h i i ) with P i ∈ M, then P i ∈ {M 2 , M 2 }, (h i = 2 or h i is of the form 3 • 2 n i -1) and P, P ∈ {M 1 , M 3 } (see Table 2.1). □ Lemma 2.6. If A is perfect, then A = A 1 .
Proof. We claim that A 2 = 1. Let P j ̸ ∈ M and Q i ∈ M. Then, P j divides neither σ(x a ), σ((x + 1) b ) nor σ(Q h i i ). Thus gcd(P

h j j , σ(A 1 )) = 1. Observe that P h j j divides σ(A 2 ) because P h j j divides A = σ(A) = σ(A 1 )σ(A 2 ). Hence, A 2 divides σ(A 2 ). So, A 2 is perfect and it is equal to 1, A being indecomposable. □ Proposition 2.1. If A 1 is perfect, then h j = 2 n j -1 for any P j ∈ M.
Proof. We refer to Table 2

.1. (i) Suppose that P j ̸ ∈ {M 2 , M 2 }. If h j is even, then σ(P h j j ) is divisible by a non- Mersenne prime. It contradicts Lemma 2.1. If hj = 2 n j u j -1 with u j ≥ 3 odd, then σ(P h j j ) = (1 +P j ) 2 n j -1 • (1+ P j + • • • + P u j -1 j ) 2 n j . Since 1+ P j + • • • + P u j -1 j = σ(P j u j -1
) is divisible by a non-Mersenne prime, we also get a contradiction to Lemma 2.1. (ii) If P j ∈ {M 2 , M 2 } and (h j is even or it is of the form 2 n j u j -1, with u j ≥ 3 odd and n j ≥ 1), then Corollary 2.1 implies that there exists ℓ ∈ N * such that M 2 ℓ ∥A 1 and

M 2 ℓ ∥A 1 . Recall that σ(M 2 2 ) = M 1 M 3 and σ(M 2 2 ) = M 1 M 3 .
We proceed as in the proof of Corollary 2.1. It suffices to distinguish four cases which give contradictions.

• Case 1:

a = 7 • 2 n -1 and b = 7 • 2 m -1 One has ℓ = 2 n + 2 m and neither M 1 nor M 3 divides σ(x a ) σ((x + 1) b ). If h j is even, then h j = 2 = ℓ. So, n = m = 0, M 1 2 ∥σ(A 1 ) = A 1 . It contradicts the part (i) of our proof. If h j = 2 n j u j -1 with u j ≥ 3 odd and n j ≥ 1, then u j = 3 and M 1 2•2 n j ∥A 1 . • Case 2: a = 7 • 2 n -1 and b = 5 • 2 m -1 One has ℓ = 2 n and M 1 ∤ σ(x a )σ((x + 1) b ). If h j is even, then 2 n = ℓ = h j = 2. So, n = 1 and M 1 2 ∥A 1 . If h j = 2 n j u j -1
, with u j ≥ 3 odd and n j ≥ 1, then u j = 3 and

2 n = ℓ = h j = 3 • 2 n j -1. It is impossible. • Case 3: a = 7 • 2 n -1, b = 3 • 2 m -1 and h j is even As above, 2 n = ℓ = h j = 2, M 1 2 m divides σ((x + 1) b ) and M 1 2 n +2 m divides σ(A 1 ) = A 1 . So, n = 1 and M 1 2 m +2 ∥A 1 . Thus, the part (i) implies that m = 0. Hence, A 1 = S 1 = x 13 (x + 1) 2 M 3 1 M 2 2 M 2 2 M 3 M 3 which is not perfect (see Example 2.1). • Case 4: a = 7 • 2 n -1, b = 3 • 2 m -1, h j = 2 n j u j -1, u j ≥ 3 odd, n j ≥ 1 One has u j = 3 and 2 n = ℓ = h j = 3 • 2 n j -1. It is impossible. □ Lemma 2.6, Proposition 2.1 and Lemma 2.3 imply Corollary 2.2. If A is perfect, then A = A 1 ∈ P. 3. Proof of Theorem 1.2
As in Section 2, we fix:

A = x a (x + 1) b i∈I P h i i = A 1 A 2 ,
where a, b, h i ∈ N, P i is a Mersenne prime,

A 1 = x a (x + 1) b P i ∈M P h i i and A 2 = P j ̸ ∈M P h j j
Sufficiencies are obtained by direct computations. For the necessities, we shall apply Lemma 3.6 and Proposition 3.1.

Lemma 3.1.

If A is unitary perfect, then σ * (x a ), σ * ((x + 1) b ), σ * (P i h i ), for any i ∈ I, are only divisible by x, x + 1 or by Mersenne primes.

Proof. Since σ * is multiplicative, σ * (A) = σ * (x a )σ * ((x + 1) b ) i∈I σ * (P i h i ). Any divisor of σ * (x a ), σ * ((x + 1) b ), σ * (P i h i ) divides σ * (A) = A. □ Lemma 3.2 ([4], Lemma 2)
. A polynomial S is unitary perfect if and only if for any irreducible polynomial P and for any m 1 , m 2 ∈ N * , we have:

(P m 1 ∥S, P m 2 ∥σ * (S)) ⇒ m 1 = m 2 ).
Example 3.1 (useful for Proposition 3.1). The polynomial

S 2 = x 14 (x+1) 7 M 1 2 M 2 3 M 2 3 M 3 M 3 is
a σ * (x a ) 3 • 2 n (x + 1) 2 n M 1 2 n 5 • 2 n (x + 1) 2 n M 3 2 n 7 • 2 n (x + 1) 2 n M 2 2 n M 2 2 n b σ * ((x + 1) b ) 3 • 2 m x 2 m M 1 2 m 5 • 2 m x 2 m M 3 2 m 7 • 2 m x 2 m M 2 2 m M 2 2 m P i h i σ * (P i h i ) M 2 3 • 2 n i (1 + M 2 ) 2 n i M 1 2 n i M 3 2 n i M 2 3 • 2 n i (1 + M 2 ) 2 n i M 1 2 n i M 3 2 n i
(i) If A is u.p, then A = A 1 . (ii) If A 1 is u.p, then h j = 2 n j for any P j ∈ M. (iii) If A is u.p, then A or A is of the form B 2 n , where B ∈ P u .
Proof. The proof of (i) is analogous to that of Lemma 2.6. The statement (iii) follows from (i), (ii) and Lemma 3.6. We only sketch the proof of (ii). Set h j = 2 n j u j , where u j is odd and n j ≥ 0.

-Suppose that P j ̸ ∈ {M 2 , M 2 }. If u j ≥ 3, then σ(P

u j -1 j
) and thus σ * (P h j j ) are divisible by a non-Mersenne prime. It contradicts Lemma 2.1.

-If P j ∈ {M 2 , M 2 } and if u j ≥ 3, then u j = 3 and (a or b is of the form 7

• 2 n ). Recall that σ * (M 2 3 ) = (1 + M 2 )M 1 M 3 and σ * (M 2 3 ) = (1 + M 2 )M 1 M 3 .
We consider two cases. The first gives non unitary perfect polynomials whereas the second leads to a contradiction.

• Case 1:

a = 7 • 2 n and b = 7 • 2 m , with n, m ≥ 0 One has M 2 ℓ ∥A 1 and M 2 ℓ ∥A 1 , with ℓ = 2 n + 2 m . Neither M 1 nor M 3 divides σ(x a ) σ((x + 1) b ). Thus, 3 • 2 n j = h j = ℓ = 2 n + 2 m . So, (n = m + 1 and n j = m) or (m = n + 1 and n j = n). Therefore, (M 1 2 ) 2 n j , M 3 2 n j and M 3 2 n j divide σ * (M h j 2 )σ * (M 2 h j ) and they divide σ * (A 1 ) = A 1 . Thus, A 1 = S 2 2 m or A 1 = S 2 2 n
where We mainly prove it by contradiction (to Corollary 4.1). Lemma 4.1 states that σ(M 2h ) is square-free, for any h ∈ N * . Recall that we set

S 2 = x 14 (x + 1) 7 M 1 2 M 2 3 M 2 3 M 3 M 3 . In both cases, A 1 is not unitary perfect because S 2 is not u.p (Example 3.1). • Case 2: a = 7 • 2 n and (b = 5 • 2 m or b = 3 • 2 m ), with n, m ≥ 0 One has ℓ = 2 n . So, we get the contradiction: 3 • 2 n j = h j = ℓ = 2 n . □ 4
M = x a (x + 1) b + 1, U 2h = σ(σ(M 2h )) and (4.1) σ(M 2h ) = j∈J P j , P j = 1 + x a j (x + 1) b j irreducible, P i ̸ = P j if i ̸ = j.
By Lemma 4.3, if there exists a prime divisor p of 2h + 1 such that σ(M p-1 ) is divisible by a non-Mersenne prime, then σ(M 2h ) is also divisible by a non-Mersenne. Therefore, it suffices to consider that 2h + 1 = p is a prime number, except for p = 3 with M ∈ {M 2 , M 2 } (see Section 4.3). The lemma below generalizes Lemma 4.10 in [START_REF] Gallardo | On Mersenne polynomials over F 2[END_REF] (with an analogous proof).

Lemma 4.3. If k is a divisor (prime or not) of 2h + 1, then σ(M k-1 ) divides σ(M 2h ).
We sometimes apply Lemmas 4.4 and 4.5 without explicit mentions.

Lemma 4.4. Let S ∈ F 2 [x] be such that s = deg(S) ≥ 1 and l, t, r, r 1 , . . . , r k ∈ N be such that r 1 > • • • > r k , t ≤ k, r 1 -r t ≤ l ≤ r ≤ s. Then (i) α l [(x r 1 + • • • + x r k )S] = α l (S) + α l-(r 1 -r 2 ) (S) + • • • + α l-(r 1 -rt) (S).
(ii) α l (σ(S)) = α l (S) if any divisor of S has degree at least r + 1.

Proof. The equality in (i) (resp. in (ii)) follows from the definition of α l (resp. from the fact: σ(S) = S + T , where deg(T ) ≤ deg(S) -r -1). □ (ii) The polynomial U 2h splits (over F 2 ) and it is a square.

(iii) The polynomial σ(M 2h ) is reducible.

Proof. (i) See [10, Corollary 4.9]. For (ii), Assumption (4.1) implies that

U 2h = σ(σ(M 2h )) = σ( j∈J P j ).
Hence, U 2h = j∈J x a j (x + 1) b j = x u (x + 1) v , where u and v are both even.

(iii) If σ(M 2h ) = Q is irreducible, then U 2h = 1 + Q is not a square. □ Lemma 4.5. One has α l (σ(M 2h )) = α l (M 2h ) if l ≤ a + b -1 and α l (σ(M 2h )) = α l (M 2h + M 2h-1 ) if a + b ≤ l ≤ 2(a + b) -1. Proof. Since σ(M 2h ) = M 2h + M 2h-1 + T , with deg(T ) ≤ (a + b)(2h -2) = 2h(a + b) -2(a + b), Lemma 4.4-(ii) implies that α l (σ(M 2h )) = α l (M 2h ) if l ≤ a + b -1 and α l (σ(M 2h )) = α l (M 2h + M 2h-1 ) if a + b ≤ l ≤ 2(a + b) -1. □ Lemma 4.6. Denote by N 2 (m) the number of irreducible polynomials over F 2 , of degree m ≥ 1. Then (i) N 2 (m) ≥ 2 m -2(2 m/2 -1) m , (ii) φ(m) < N 2 (m) if m ≥ 4
, where φ is the Euler totient function,

(iii) For each m ≥ 4, there exists an irreducible polynomial of degree m, which is not a Mersenne prime.

Proof. (i) See [START_REF] Lidl | Finite Fields, Encyclopedia of Mathematics and its applications[END_REF], Exercise 3.27, p. 142.

(ii) If m ∈ {4, 5}, then direct computations give φ(4) = 2, N 2 (4) = 3 and φ(5) = 4, N 2 (5) = 6. Now, suppose that m ≥ 6. Consider the function f (x) = 2 x -2(2 x/2 -1) -x 2 , for x ≥ 6. The derivative of f is a positive function. So, f (x) ≥ f (6) > 0 and Let q = 2 r -1 be a Mersenne prime number. Then, any irreducible polynomial P of degree r is primitive. In particular, each root β of P is a primitive element of the field F 2 r , so that β is of order q in F 2 r \ {0}. Lemma 4.9. Let P i = 1+x a i (x+1) b i be a prime divisor of σ(M p-1 ), where 2 a i +b i -1 = p i is a prime number. Then, p i = p and σ(M p-1 ) is divisible by any irreducible polynomial of degree a i + b i . Furthermore, at least one of those divisors is not a Mersenne prime if a i + b i ≥ 4.

x < 2 x -2(2 x/2 -1) x . Thus, φ(m) ≤ m < 2 m -2(2 m/2 -1) m ≤ N 2 (m).
Proof. The polynomial P i is primitive. If α is a root of P i , then (M p + 1)(α) = 0 and M (α) = α r for some 1 ≤ r ≤ p i -1. Thus, 1 = M (α) p = α rp , with ord(α) = p i . So, p i divides rp and p i = p. Any irreducible polynomial S of degree a i + b i is primitive. Let β be a root of S. One has ord(β) = p i = p, S(β) = 0 and M (β) = β s , for some 1 ≤ s ≤ p i -1. Thus, M (β) p = β ps = 1 and S divides M p + 1 = x a (x + 1) b σ(M p-1 ). The third statement follows from Lemma 4.6-(iii).

□ Corollary 4.2. For any

i ∈ J, a i + b i ≤ 3 or 2 a i +b i -1 is not prime. Lemma 4.10. Let P, Q ∈ F 2 [x] be such that deg(P ) = r, 2 r -1 is prime, P ∤ Q(Q+1) but P | Q p + 1. Then 2 r -1 = p. Proof. The polynomial P is primitive. If β is a root of P , then ord(β) = 2 r -1. Moreover, Q(β) ̸ ∈ {0, 1} because P ∤ Q(Q + 1). Thus, Q(β) = β t for some 1 ≤ t ≤ 2 r -2. Hence, 1 = Q(β) p = β tp . So, 2 r -1 divides tp and 2 r -1 = p.
□ Corollary 4.3. Let r ∈ N * be such that 2 r -1 is a prime distinct from p. Then, no irreducible polynomial of degree r divides σ(M p-1 ).

Proof. If P is a prime divisor of σ(M p-1 ) with deg(P ) = r, then P divides M p + 1 and by taking Q = M in the above lemma, we get a contradiction. □

In the following lemma and two corollaries, we suppose that p is a Mersenne prime of the form 2 m -1 (with m prime). Lemma 4.11. Let P, Q ∈ F 2 [x] be such that P is irreducible of degree m and P ∤

Q(Q + 1). Then, P divides Q p + 1. Proof. The polynomial P is primitive. If β is a root of P , then ord(β) = 2 m -1 = p, Q(β) ̸ ∈ {0, 1} because P ∤ Q(Q + 1). Thus, Q(β) = β t for some 1 ≤ t ≤ p -1. Hence, Q(β) p = β tp = 1. So, P divides Q p + 1.
□ Corollary 4.4. Any irreducible polynomial P ̸ = M (Mersenne or not), of degree m, divides σ(M p-1 ).

Proof. We may apply Lemma 4.11, with Q = M , because P does not divide Proof. For (i), (iii) and (iv), use [START_REF] Gallardo | Characterization of Sporadic perfect polynomials over F 2[END_REF], Lemmas 5.9, 5.10, 5.15 and 5.17 Proof. The polynomial U 2 is a square, so 0 = α 1 (U 2 ) = α 1 (B) + 1 and thus α 1 (B) = 1. Lemma 4.19-(iii) implies that 0 = α 3 (σ(M 2 )) = α 3 (B) + α 2 (B) + α 1 (B). Therefore, α 3 (U 2 ) = α 3 (B) + α 2 (B) = α 1 (B) = 1. □ Remark 4.1. Our method fails for p = 7. Indeed, for many M , one has α 3 (U 6 ) = α 5 (U 6 ) = 0. So, we do not reach a contradiction. We should find a large enough odd integer l such that, α l (U 6 ) = 0. But, this does not appear always possible.

x a (x + 1) b M = M (M +1) = Q(Q+1). So, P is odd and it divides M p +1 = (M +1) σ(M p-1 ) = x a (x + 1) b σ(M p-1 ). □ Corollary

Proposition 1 . 1 .

 11 Let h ∈ N * and let M ∈ F 2 [x] be a Mersenne prime. Then in the following cases, σ(M 2h ) is divisible by a non-Mersenne prime:

Lemma 3 . 4 .Definition 3 . 1 .Lemma 3 . 5 .

 343135 Let C ∈ F 2 [x] \ {0, 1} be u.p. Then C is even, C and C 2 r are also u.p, for any r ∈ N.Proof. If D is a divisor of C, then D divides C and D 2 r divides C 2 r . Thus, σ * (C) = σ * (C) = C and σ * (C 2 r ) = (σ * (C)) 2 r = C 2r . It remains to prove that C is even. Consider an irreducible divisor P of C and k ∈ N * such that P k ∥C. The polynomial 1 + P is even and divides 1 + P k = σ * (P k ). So, 1 + P divides σ * (C) = C. □ We denote by ∼ the relation on F 2 [x] defined as: S ∼ T if there exists ℓ ∈ Z such that S = T 2 ℓ . ([3], Section 2) The relation ∼ is an equivalence relation on F 2 [x]. Each equivalence class contains a unique polynomial B which is not a square, with val x (B) ≤ val x+1 (B).

Lemma 3 . 6 (

 36 [START_REF] Gallardo | On even (unitary) perfect polynomials over F 2[END_REF], Theorem 1.3). If h i = 2 n i for any i ∈ I, then A (or A) is of the form B 2 n , where B ∈ P u . Proposition 3.1.

.

  Proof of Proposition 1.1 That proposition partially solves Conjecture 4.1. ([10], Conjecture 1.1) Let h ∈ N * and let M ∈ F 2 [x] be a Mersenne prime. Then, σ(M 2h ) is always divisible by a non-Mersenne prime, except for M ∈ {M 2 , M 3 } and h = 1.

4. 1 .

 1 Useful facts. For S ∈ F 2 [x]\{0, 1}, of degree s, we denote by α l (S) the coefficient of x s-l in S, 0 ≤ l ≤ s. One has: α 0 (S) = 1. Lemma 4.1 ([10], Lemmas 4.6 and 4.8). The polynomial σ(M 2h ) is square-free and M ̸ = M 1 .Lemma 4.2 ([10], Theorem 1.4). Let h ∈ N * be such that p = 2h + 1 is prime and let M be a Mersenne prime such that M ̸ ∈ {M 2 , M 2 } and ω(σ(M 2h )) = 2. Then, σ(M 2h ) is divisible by a non-Mersenne prime.

Corollary 4. 1 .

 1 (i) The integers u = j∈J a j and v = j∈J b j are both even.

( 4 . 7 .

 47 iii) We remark that if 1 + x c (x + 1) d is a Mersenne prime, then gcd(c, d) = 1. So, gcd(c, c + d) = 1. Therefore, the set M m of Mersenne primes of degree m is a subset of {x c (x + 1) m-c + 1 : 1 ≤ c ≤ m, gcd(c, m) = 1}. Thus, #M m ≤ #{c : 1 ≤ c ≤ m, gcd(c, m) = 1} = φ(m).Hence, there exist at least N 2 (m) -φ(m) irreducible non-Mersenne polynomials, with N 2 (m) -φ(m) ≥ 1, by (ii).□ Lemma For any j ∈ J, ord p (2) divides a j + b j = deg(P j ). Proof. Set d = gcd i∈J (a i + b i ). By Lemma 4.13 in [10], p divides 2 d -1. Thus, ord p (2) divides d. □ Lemma 4.8. ([11], Chap. 2 and 3)

4 . 5 . 4 . 2 .Lemma 4 . 12 .

 4542412 The polynomial M 1 (resp. M 2 , M 2 ) divides σ(M p-1 ) if and only if (M ̸ = M 1 and p = 3) (resp. M ̸ = M 2 and p = 7, M ̸ = M 2 and p = 7). Proof. Apply Corollary 4.4 with m ∈ {2, 3}. □ In order to carry on the proof (of Proposition 1.1), we distinguish three cases. Case I: M ∈ {M 1 , M 3 , M 3 }. Lemma 4.1 implies that M ̸ = M 1 . It suffices to suppose that M = M 3 . We refer to Section 5.2 in [9]. Put D = M 1 M 2 M 2 . By [9, Lemma 5.4], we have to consider four situations: (i) gcd(σ(M 2h ), D) = 1, (ii) σ(M 2h ) = M 1 B, with gcd(B, D) = 1, (iii) σ(M 2h ) = M 2 M 2 B, with gcd(B, D) = 1, (iv) σ(M 2h ) = DB, with gcd(B, D) = 1, where any irreducible divisor of B has degree exceeding 5. The following lemma contradicts the fact that U 2h is a square. One has α 3 (U 2h ) = 1 or α 5 (U 2h ) = 1.

α 1 (□Corollary 4 . 9 .

 149 . (ii) Since σ(M 2h ) = (x 2 + x + 1)B and U 2h = (x 2 + x)σ(B), we obtain (by Lemmas41 (M 2h ) = α 1 (σ(M 2h )) = α 1 (B) + 1, α 3 (U 2h ) = α 3 (σ(B)) + α 2 (σ(B)) = α 3 (B) + α 2 (B), 0 = α 3 (M 2h ) = α 3 (σ(M 2h )) = α 3 (B) + α 2 (B) + α 1 (B). Thus, α 3 (U 2h ) = α 3 (B) + α 2 (B) = α 1 (B) = 1. □ Lemma 4.20. Some coefficients of U 2 and B satisfy:α 1 (U 2 ) = α 1 (B) + 1, α 2 (U 2 ) = α 2 (B) + α 1 (B), α 3 (U 2 ) = α 3 (B) + α 2 (B). Proof. Corollary 4.8 implies that U 2 = σ(σ(M 2 )) = σ((1 + x + x 2 )B) = σ(1 + x + x 2 )σ(B) = (x 2 + x)σ(B).Any irreducible divisor of B has degree more than 3. Hence,α l (σ(B)) = α l (B), for 1 ≤ l ≤ 3. U 2 ) = α 1 (σ(B)) + 1 = α 1 (B) + 1, α 2 (U 2 ) = α 2 (σ(B)) + α 1 (σ(B)) = α 2 (B) + α 1 (B), α 3 (U 2 ) = α 3 (σ(B)) + α 2 (σ(B)) = α 3 (B) + α 2 (B). The coefficient α 3 (U 2 ) equals 1.

Table 3 .

 3 1.

	The following table, obtained from Lemmas 2.1, 2.4 and 3.3, are useful to prove
	Proposition 3.1.

not unitary perfect since x 14 ∥S 2 and x 10 ∥σ * (S 2 ). Similar arguments give Proposition 3.1 which finishes our proof. Lemma 3.3. Let S ∈ F 2 [x] be an irreducible polynomial. Then, for any n, u ∈ N with u odd, σ * (S 2 n u ) = (1 + S) 2 n (σ(S u-1 )) 2 n .

Univ. Brest Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, 6, Av. Le Gorgeu, C.S. 93837,

Brest Cedex 3, France. Email address: luis.gallardo@univ-brest.fr Email address: olivier.rahavandrainy@univ-brest.fr

Lemma 4.14. For p ∈ {5, 7}, some non-Mersenne prime divides σ(M p-1 ).

Proof. Here, h ∈ {2, 3}. By direct computations, U 4 = x 3 (x + 1) 6 (x 3 + x + 1) and U 6 = x 8 (x + 1) 4 (x 3 + x + 1) 2 which do not split (despite that U 6 is a square). □ 4.3.2. II-2: 2h + 1 = 3 w , for some w ≥ 2. In this case, 9 divides 2h + 1 and σ(M 8 ) divides σ(M 2h ) (by Lemma 4.3). But, σ(M 8 ) = (x 2 + x + 1)(x 4 + x 3 + 1)(x 6 + x + 1)(x 12 + x 8 + x 7 + x 4 + 1), where x 6 + x + 1 = 1 + x(x + 1)M 3 is not a Mersenne prime.

II-3:

2h + 1 is (divisible by) a prime p ̸ ∈ {3, 5, 7}. We may write p = 2h + 1 with h ≥ 4.

Lemma 4.15.

). Hence, we get (ii) and (iii). □ Corollary 4.6. The coefficient α 3 (U 2h ) equals 1.

Proof. The previous lemma implies that We shall prove that α 3 (U 2 ) = 1 (Corollary 4.9), a contradiction to the fact that U 2 is a square. Corollary 4.5 gives Lemma 4.18. (i) The trinomial 1 + x + x 2 divides σ(M 2 ). (ii) No irreducible polynomial of degree r ≥ 3 such that 2 r -1 is prime, divides σ(M 2 ). Proof. We directly get (i) and (ii). For (iii), σ(M 2 ) = 1 + M + M 2 = x 2a (x + 1) 2b + x a (x + 1) b + 1. Moreover, 2a + 2b -3 > a + b because a + b ≥ 4 and x 2a (x + 1) 2b is a square. So, α 3 (σ(M 2 )) = α 3 (x 2a (x + 1) 2b ) = 0. □


Stéphane Chaillou

Harnessing the microbial potential of fermented food for healthy & sustainable food systems

INTRODUCTION

• Demonstrating the health benefit of fermented foods-based diet.

• Designing microbial consortia for plant-based fermented foods.

• Involving citizens and food system actors in several EU living labs.

What is DOMINO?

X1 X2 X3 X4 X5 N1 +1 +1 -1 -1 +1 N2 -1 +1 +1 +1 -1 N3 -1 -1 +1 -1 +1 N4 +1 -1 -1 +1 -1 N5 -1 +1 +1 +1 -1 N6 +1 -1 -1 -1 +1

General concept of DOMINO 1 •

 1 Coordinated by INRAE.• kick-off 1 st March 2023 • 5 years (2023-2028) and budget of 12 M€ • 19 partners in 10 EU countries • Over 100 scientists in more than 8 scientific disciplines • Over 10,000 of fermented food; animal & human gut 'omics' data 2 DOMINO summary in a few numbers About the project 3

Background••

  downstream process, fermented food can be with or without living micro-organisms Plant-based fermented food can be of major importance in the transition from an animal-to more plant-based diet, and in so doing to provide sustainability benefits A tiny fraction of the large natural diversity of food microbial strains and consortia are currently harnessed the DOMINO effect The concept and scientific objectives of DOMINO 6 Focusing on fermented food-related health biomarkers • Combining of health models (Human, animals and ex-vivo models) Setting synthetic ecology approaches for knowledge-driven design of microbial solutions • Co-creating plant-based ferm. food prototypes (from traditional & new plant-based products) • Evaluating market opportunities & consumer acceptance • Putting computational biology at the heart of knowledge acquisition • Opening the data and tools for the scientific community DOMINO from objectives t impacts 1. Health assessment of fermented food based diet MATCHING PATTERN OF MICROBIOMES, PHYSIOLOGY AND EX-VIVO DATA TO BETTER FOCUS ON HEALTH BIOMARKERS LINKED TO FERMENTED

  Figure.2

  data (public & those generated within the project) Metagenomic & MAGs data (public & those generated within the project)


Elwan Héry

Philippe Xu

Philippe Bonnifait

A Study of Different Observation Models for Cooperative Localization in Platoons

Localization and perception for autonomous vehicles are often studied separately. However, they can be regroup on a dynamic map representing the environment of the vehicle. This dynamic map can be exchanged with other vehicles to be fused with their own dynamic maps to increase their accuracy. This paper presents a decentralized data fusion method for cooperative localization based on both Extended Kalman Filter and Covariance Intersection Filter. Different observation models of the relative measurements from the perception (Cartesian and polar relative poses, distances, bearings and relative yaws) are compared. The approach is tested on data for 10 vehicles generated from a real dataset and completed with a simulated perception.

I. INTRODUCTION

Localization and perception are among the bigger challenges for autonomous vehicle's navigation. The ego-vehicle needs to be correctly localized, but also understand where the other vehicles around it are. To extend the perception range outside the Field of View (FoV) of the ego-vehicle, wireless communications between vehicles can be used. To improve the localization and perception at the same time, the information coming from localization, perception and communication can be fused to obtain a more accurate representation of the scene surrounding the ego-vehicle.

One of the first methods to solve cooperative localization issues, was to see the robot's formation as a rigid structure [START_REF] Ahmad | Cooperative robot localization and target tracking based on least squares minimization[END_REF]. The robots are nodes of the rigid graph and the perception measurements such as bearing [START_REF] Zelazo | Rigidity Theory in SE(2) for Unscaled Relative Position Estimation using only Bearing Measurements[END_REF] or range [START_REF] Zelazo | Decentralized Rigidity Maintenance Control with Range Measurements for Multi-Robot Systems[END_REF], [START_REF] Hoangt | Cooperative Localization in VANETs: An Experimental Proof-of-Concept Combining GPS, IR-UWB Ranging and V2V Communications[END_REF] correspond to the links. Graph optimization methods can then be applied to solve the cooperative localization issue. Another solution is to use filtering techniques on a global state composed of the states of every vehicle. An Extended Kalman Filter (EKF) can be used on each vehicle to obtain a decentralized approach.

When using a decentralized approach with communication, if the states are exchanged several times, these observations and the state of the ego-vehicle have correlated errors. This issue named data incest makes the EKF over-converge and the estimated state not consistent anymore. Different solutions exist, only raw measurements can be communicated, such as GNSS (Global Navigation Satellite System) pseudoranges [START_REF] Lassoued | Mutual Localization and Positioning of Vehicles Sharing GNSS Pseudoranges: Sequential Bayesian Approach and Experiments[END_REF] instead of the already fused states. Two EKF [START_REF] Aufrère | A state exchange approach in real conditions for multi-robot cooperative localization[END_REF], [START_REF] Karam | Localization of a Group of Communicating Vehicles by State Exchange[END_REF] can also be used. One without the information of other vehicles (it gives the estimation of the state for the 1 Nantes Université, École Centrale de Nantes and CNRS LS2N, 44300 Nantes, France first-name.last-name@ec-nantes.fr 2 Université de technologie de Compiègne, CNRS, Heudiasyc UMR 7253, France firstname.surname@hds.utc.fr other vehicles) and one with this information for the egovehicle. To avoid the data incest issue, other filters exist, such as the Covariance Intersection Filter (CIF) [START_REF] Julier | A non-divergent estimation algorithm in the presence of unknown correlations[END_REF]. The uncertainty of the estimation of this filter is often pessimistic. A solution to obtain a compromise between the EKF and the CIF is to use the Split Covariance Intersection Filter (SCIF) [START_REF]General Decentralized Data Fusion with Covariance Intersection (CI)[END_REF]. This filter has been used several times for cooperative localization [START_REF] Li | Cooperative multi-vehicle localization using split covariance intersection filter[END_REF], [START_REF] Wanasinghe | Decentralized Cooperative Localization for Heterogeneous Multi-robot System Using Split Covariance Intersection Filter[END_REF] as well as for cooperative perception [START_REF] Lima | Data Fusion with Split Covariance Intersection for Cooperative Perception[END_REF]. For cooperative perception [START_REF] Caillot | Survey on Cooperative Perception in an Automotive Context[END_REF], deep learning fusion method exists with early, intermediate or late fusion [START_REF] Ren | Collaborative Perception for Autonomous Driving: Current Status and Future Trend[END_REF], where raw data, feature maps or already detected objects are fused.

The cooperative localization data fusion algorithm presented in this work is based on multiple filters: an EKF for GNSS poses, kinematics (longitudinal speeds and yaw rates) and perception sensors observations and CIF for vehicles states received through communication. Both filters update the same state of the dynamic map of the ego-vehicle. In this paper, the main contributions are:

• The comparison of different observation models (Cartesian and polar relative poses, distances, bearings and relative yaws) for the relative measurements of the perception to improve the state of the vehicles both in terms of accuracy and consistency. • The evaluation of the method on a hybrid simulation generated from a real dataset and completed with simulations to obtain a platoon of 10 vehicles. The perceptions are simulated with a FoV taking into account vehicles obstructions. Compared to our previous work [START_REF] Héry | Consistent decentralized cooperative localization for autonomous vehicles using LiDAR, GNSS, and HD maps[END_REF] the goal is, therefore, to study different observation models on a much longer platoon (10 vehicles instead of 2).

In Sec. II, the data fusion algorithm is presented globally and with its different steps: EKF prediction, EKF updates with the different observations models and CIF updates from the states of other vehicles. After this, in Sec. III the experimental and simulated setup is described before analyzing the obtained results.

II. DATA FUSION ARCHITECTURE

A. Global Architecture

The data fusion is organized with a decentralized architecture as shown in Fig. 1. Each vehicle estimates its state (its position, yaw, longitudinal speed and yaw rate) and the state of every vehicle of the platoon (composed of the same elements) as described in Tab. I. Each vehicle has, therefore, a global state of all the dynamics agents (only We suppose in this work that all the observations are synchronized. As the data used for the evaluation of our approach will be partially simulated to obtain more vehicles, this assumption greatly simplified the architecture. A more complex asynchronous cooperative localization architecture was studied in [START_REF] Héry | Consistent decentralized cooperative localization for autonomous vehicles using LiDAR, GNSS, and HD maps[END_REF]. It is not in the scope of this work. This choice was made to obtain a repeatable algorithm that loads the observations one after the other to obtain results that will not change, if the computation time is slightly different. Indeed, if each EKF update is done in a different thread, depending on the computation time of each thread, the different updates would be applied with a different order, which will give different results. This was one of the main issue of [START_REF] Héry | Consistent decentralized cooperative localization for autonomous vehicles using LiDAR, GNSS, and HD maps[END_REF] (much realistic but also less easy to study the results). A linear architecture, as shown in this work, gives repeatable results that can be easily compared.

In Fig. 1, one can see the different steps to update the state of each vehicle with the different observations. For each time step, the state is first predicted at the new time step with an EKF evolution model. After this, the state is updated with an EKF with the observation model of the kinematics (longitudinal speed and yaw rate) of the GNSS pose and of every perceived vehicle. Each relative measurement of the perception is taken into account one after the other on different EKF updates. We choose to keep sequential updates of the observations instead to fuse all the synchronized observations on one update, as it would be the case on an asynchronous architecture. This choice gives a more modular architecture, able to work even if one observation is missing, without complexification of the architecture, as it would be the case in an asynchronous architecture. Finally, the communication between vehicles is taken into account with a Covariance Intersection Filter (CIF) to avoid data incest.

B. EKF prediction

The first step is to predict the state of the dynamic map at the new time step. The following evolution model is used:

X t k = evo(X t k-1 ) = X t k-1 + V t k-1 (1)
For the vehicle i of the dynamic map of the ego-vehicle:

            . . . x i y i θ i v i ω i . . .             t k =             . . . x i y i θ i v i ω i . . .             t k-1 +             . . . v i ∆t cos(θ i + ∆t 2 ω i ) v i ∆t sin(θ i + ∆t 2 ω i ) ω i ∆t 0 0 . . .             t k-1 (2)
All elements of the arrays are at the time of the indices of the arrays. The dynamic map X t k of the ego-vehicle at the time t k of the step k is obtained from the dynamic map X t k-1 of the ego-vehicle at the time t k-1 from this previous step and the motion V t k-1 of every vehicle computed from the longitudinal speeds, yaw rates and yaws at the previous time step t k-1 and with the time delay ∆t = t k -t k-1 . To compute the new covariance matrix for the dynamic map, we use the following prediction model:

Σ Xt k = J evo Σ Xt k-1 J T evo + Σ evo ∆t (3)
where J evo is the Jacobian matrix of the evolution model evo(X i,t k-1 ) previously defined and Σ evo the covariance matrix of the evolution model noise.

C. EKF updates

Once the state of the dynamic map of the vehicle is at the new time, we can update it with the different observations. To simplify the notations we remove the time t k , as the state and the observations are all at this time in the rest of the paper. To update the dynamic map of the ego-vehicle, we first compute the innovation from the observation Z and the observation model obs(X). The covariance of the innovation is also estimated from the Jacobian matrix of the observation model J obs and the covariance matrix of the observation noise

Σ Z : Y = Z -obs(X) (4)
Σ Y = J obs Σ X J T obs + Σ Z (5)
Then, the Kalman gain is computed: Finally, the state and the covariance matrix are updated:

K = Σ X J T obs Σ -1 Y (6)
X = X + KY (7)
Σ X = (I -KJ obs )Σ X (I -KJ obs ) T + KΣ Z K T (8)
The update of the EKF is used for all the following observations: we note j z i the variable z of vehicle i (with the notation "ego" for the ego-vehicle) in the frame of vehicle j (this exponent is removed if it is in the global frame).

1) Kinematics: The kinematics containing the longitudinal speed and yaw rate with the observation model are:

obs vel (X) = v ego ω ego (9)
2) GNSS pose: The pose computed by the GNSS receiver is updated with the observation model:

obs pose (X) =   x ego y ego θ ego   (10)
3) Relative measurement of the perception: For each perceived vehicle i, detected by the ego-vehicle, we can define a different observation model of the relative pose ego q i of this vehicle i in the mobile frame of the ego-vehicle:

obs pose cartesian (X) = ego q i =   ego x i ego y i ego θ i   = R ego (q i -q ego ) (11)
with the rotation matrix:

R ego =   cos θ ego sin θ ego 0 -sin θ ego cos θ ego 0 0 0 1   (12)
However, we showed in [START_REF]Pose and covariance matrix propagation issues in cooperative localization with LiDAR perception[END_REF] the issues of propagating a pose with uncertainty. The obtained distribution at the end of the transformation is not a Gaussian distribution anymore but a banana shaped one, illustrated in Fig. 2 (the uncertainties are exaggerated in this figure to better observe this shape). In this filter, the observation model obs pose cartesian (X) gives an uncertainty distribution like this. To better understand this shape, another solution is to represent the observation by a polar relative pose. With this new observation, we have the following observation model:

obs pose polar (X) =   ego r i ego α i ego θ i   =   (x i -x ego ) 2 + (y i -y ego ) 2 arctan 2(y i -y ego , x i -x ego ) -θ ego θ i -θ ego   (13)
One can see in Fig. 2, that the non-linearity creating a banana shaped distribution instead of Gaussian distribution, are linked to the yaw and relative yaw angle uncertainties. It is also possible to define observation models with only the distance ego d i to be independent of these angles or with only the bearing angle ego α i or the relative yaw ego θ i if we want to observe their effects. To obtain the associated observation model, we just have to keep the correct line of the polar observation model obs pose polar (X).

D. CIF updates

We suppose in this work that every vehicle is able to communicate its dynamic map to all the other vehicles. To use these new observations while avoiding data incest, we use a Covariance Intersection Filter (CIF) instead of an EKF update. As we supposed that each vehicle is able to communicate with every vehicle and that all vehicles are present in the same order inside each dynamic map, the observation model is the identity.

To update the dynamic map with the CIF, we first compute the innovation from the state X (the dynamic map of the ego-vehicle) and the observation Z (the dynamic map communicated by another vehicle):

Y = Z -X (14)
Then, we search the weight w ∈ [0, 1] that minimizes the following cost function:

det Σ X (15) with Σ -1 X = wΣ -1 X + (1 -w)Σ -1 Z ( 16
)
Once the optimal weight ŵ is estimated, we can update the state and its covariance matrix:

Σ = ( ŵΣ -1 X + (1 -ŵ)Σ -1 Z ) -1 (17) X = Σ X ( ŵΣ -1 X X + (1 -ŵ)Σ -1 Z Z) (18)
As this data fusion method includes an optimization, this is the step that takes the longest time in the complete data fusion architecture. Our implementation is coded in python and is not optimized. The computation time is, therefore, not the one to expect from a real-time application. However, one can see in this architecture, when N -1 vehicles can communicate with the ego-vehicle, N -1 CIF updates are required with a state and measurement of size 5 × N (as one can see on Tab. I). This means that the computation time will greatly increase when the number of communicating vehicles increases.

III. EXPERIMENTAL RESULTS

A. Experimental scenarios

The experimentation is based on real data from the dataset that we published 1 with [START_REF] Héry | Consistent decentralized cooperative localization for autonomous vehicles using LiDAR, GNSS, and HD maps[END_REF]. This dataset is composed of the sensors data of a Renault Zoe car recorded on 10 laps of a test road of the Heudiasyc lab in Compiègne, France. This road is composed of two roundabouts with a straight line between them. In this dataset, the following sensor data are used for this work:

• GNSS poses from an Ublox M8T (unbiased with a mean filter with a 30 s rolling window). • Longitudinal speeds and yaw rates computed from the odometry and IMU coming from the CAN bus of the vehicle.

• Ground truth of all the previous information from a SPAN CPT an IMU / GNSS receiver with RTK corrections. A platoon of 10 vehicles is generated from this vehicle, by replaying the same data with a different time delay for each vehicle. Finally, the relative poses from the perception between two vehicles (which cannot be generated with this method) are simulated from the ground truth using the same transformation as Eq. [START_REF] Wanasinghe | Decentralized Cooperative Localization for Heterogeneous Multi-robot System Using Split Covariance Intersection Filter[END_REF]. Gaussian noises with a standard deviation of 5 cm for the relative positions coordinates and 0.05 rad for the relative yaws are added to these relative poses. The 360°field of view (FoV) of the perception sensor of each vehicle is simulated by ray casting to take into account the obstructions of the other vehicles, as shown in Fig. 3. One can see on this figure, when the vehicles are on the roundabout, more vehicles can be detected than when the vehicles are on the straight lane. This 360°FoV is compared to a limited FoV where only the vehicle in front of the egovehicle is detected. In practice, these relative measurements could be obtained with perception sensors more accurate to estimate bearings, such as cameras, or distances, such as LiDAR. We chose a noise model of what can be expected from a LiDAR, similarly to what we observed in [START_REF]LiDAR based relative pose and covariance estimation for communicating vehicles exchanging a polygonal model of their shape[END_REF].

B. Global localization

The different results of this paper are computed with the observation models of Cartesian relative pose "Cartesian pose (front FoV)" with the limited front FoV and Cartesian relative pose "Cartesian pose (360°FoV)", polar relative pose "polar pose (360°FoV)", distance "Distance (360°FoV)", bearing "Bearing (360°FoV)" and relative yaw "Yaw (360°F oV)" for the 360°FoV. Fig. 4a shows the mean of the norms of the positions errors ϵ p and Fig. 4b, the mean of the absolute value of the yaw errors ϵ θ computed from the estimates (p and θ) with respect to the ground truth (p ref and θ ref ):

ϵ p = ||p -p ref || , ϵ θ = | θ -θ ref | (19)
We note • the mean value on all the dataset time steps. Fig. 4c shows the consistence C for a 95% probability of success of a χ 2 test:

C = (q -q ref ) T ( Σ q + Σ qref ) -1 (q -q ref ) < χ 2 3;0.05 (20)
One can see in Fig. 4a and Fig. 4b, that the perception with relative poses observations (Cartesian or polar) increase the accuracy of the position and yaw of the vehicle. This accuracy already starts to increase when only one vehicle is detected by the ego-vehicle (with the front FoV) and is even better when many vehicles can be perceived all around the ego-vehicle (with the 360°FoV). These results can be seen in particular for vehicles in the center of the platoon, as they are able to perceive more vehicles than the one in front or at the back of the platoon. One can see when using only the distances, that the position accuracy increases compare to no perception but not the yaw accuracy. When using the bearings, both the position and yaw accuracy increase, even if the position accuracy increases less than when using distances. The results for these two sub-observations could be expected from Eq. 13, as the distance component depends on the positions and the bearing depends on the positions and ego yaw. The yaw observations, however, do not have any influence on the position or yaw accuracies. One can also see that, to be able to increase correctly the position accuracy with only a limited perception like distances or bearings, it is necessary to observed multiple vehicles (like the vehicles in the center of the platoon). For every observation models, one can see on Fig. 4c that the pose is always consistent and even pessimistic. Fig. 5 also illustrates these results for the vehicle 6 on the total duration of the dataset. One can see that for every observation models, the position coordinates and yaw errors are able to stay inside their uncertainty intervals.

C. Relative localization

Fig. 6a shows the mean of the norms of the relative positions errors and Fig. 6b shows the mean of the absolute value of the relative yaw errors with respect to the ground truth. Finally, Fig. 6c shows the relative consistency for a 95% probability. They are computed similarly to the global localization for each relative pose in the frame of the vehicle 6 from its dynamic map instead of the global frame. No result is displayed for vehicle 6 as it is the ego-vehicle used to compute the relative poses. One can see that, when using relative poses observations, the relative localization is greatly improved, for the position and yaw errors. We can also see that the reduced models, such as distance or yaw models, with less information are less accurate. However, the bearing increase significantly the relative position and yaw accuracies. Even if the global localization is more accurate globally with the distance model compare to no perception, its relative localization is not improved. One can see in these results that the vehicles in the middle of the platoon are also more accurate than the one on the extremities, similarly to the global localization. This result is also present when no perception is available, as the relative poses with the vehicles near the ego-vehicle are more accurate when they are computed with the same transformation as Eq. ( 11). Similarly to the global localization, the relative localization is consistent and even pessimistic for every observation models.

IV. CONCLUSION

This paper described a distributed data fusion method for cooperative localization and perception. This approach was tested on a platoon composed of 10 vehicles not always able to see every other vehicles. The results of this work presents a comparison of different observation models (Cartesian and polar relative poses, distances, bearings and relative yaws). One can see in these results that the best models are the Cartesian or polar relative pose perception models, as they can greatly increase the global and relative localization while staying consistent. This could be expected, as these models contain more information than the others (three parameters In future works, we would like to extend the dynamic map with diverse dynamic agents (e.g., vehicles but also pedestrians or bicycles) and complete it with static information to obtain a global representation of the static and dynamic environment of the autonomous vehicle.

Fig. 1 :

 1 Fig. 1: Global architecture of the dynamic maps updates of N vehicles based on EKF predictions and updates and CIF updates. The dynamic map states are in green and the observations in blue.

Fig. 2 :

 2 Fig. 2: Polar representation with the distance d, yaw θ and bearing α and with its uncertainty represented by the banana shape and the ellipse of uncertainty approximating this distribution.

Fig. 3 :

 3 Fig. 3: The simulated field of view of vehicle 6 (with the green bounding box) detects the red bounding boxes of the other vehicles by ray casting (in orange).

1

  https://datasets.hds.utc.fr/project/2 (a) Mean position errors (in m). (b) Mean yaw errors (in °). (c) Consistency of the poses (in %).

Fig. 4 :

 4 Fig. 4: Global localization of the ego-vehicle from each dynamic map for the different observation models of the perception.

Fig. 5 :

 5 Fig. 5: Position and yaw errors of the estimated poses of vehicle 6 in its own dynamic map with their ±2σ uncertainty intervals in function of the time steps for every observation models of the perception.

  (a) Mean relative position errors (in m). (b) Mean relative yaw errors (in °). (c) Consistency of the relative poses (in %).

Fig. 6 :

 6 Fig. 6: Relative localization of each vehicle in the frame of vehicle 6 from the dynamic map of vehicle 6 for the different observation models of the perception.

TABLE I :

 I State of the vehicle dynamic map

			Dynamic map: X
	...		Vehicle i state: X i	...
		pose: q i	velocity: u i
	position: p i	yaw: θ i	speed:v i	yaw rate: w i
	x i	y i
	vehicles in our work) surrounding it. We call this global state
	a dynamic map, as it represents the dynamic environment
	of the ego-vehicle. By grouping all these elements into
	one dynamic map, each vehicle is able to obtain a high-
	level representation of the dynamic agents surrounding it.
	The localization, perception and communication information
	are fused to obtain a more accurate representation of this
	dynamic map. To model the uncertainty of the dynamic map,
	a covariance matrix will be estimated as well. In this paper,
	we use the notation Σ Z for the covariance matrix of any
	variable Z.

ACKNOWLEDGMENT

This work was carried out in the framework of the NExT Senior Talent Chair DeepCoSLAM, which weas funded by the National Agency for Research (ANR-16-IDEX-0007), and with the support of Région Pays de la Loire and Nantes Métropole.


Ramón Martí Solano

Isabel Celis Villalba

Proverbs in everyday speech versus frozen proverbs in dictionaries and texts: a corpus-based study

Keywords: Spanish proverbs, orally transmitted proverbs, variant forms, proverb modifications, semantic shift, corpus-based analysis

When proverbs are used in everyday speech, speakers tend to reproduce them with the same lexical and morphosyntactic structure and apply them to the same situations and contexts.

The aim of this paper is to highlight the differences between proverbs when used popularly in speech and their actual form and meaning in proverb dictionaries, collections or compilations, diachronic and synchronic corpora, literary quotations and the daily press.

Our research focuses on a small selection of traditional Spanish proverbs collected in Andalusia, southern Spain.

The selection includes, among others, the following proverbs: (a) No es señora quien lo es, sino quien lo sabe ser. [A lady is not she who is a lady but she who knows how to be one.] (b) No se puede luchar contra el destino, el que nace lechón muere cochino. [One cannot fight against destiny, he who is born a piglet dies a pig.] (c) Tiene el mal del milano, las patas malas y el pico sano. [He is like the sick kite, bad legs and a healthy beak.]

Sometimes proverbs are adapted according to the speaker and thus the canonical señor (gentleman) becomes señora (lady) as in (a). Other times, speakers add an extension to a single-sentence proverb which functions as a sort of down-to-earth rhymed illustration, as seen in the first part of (b). However, and most often, lexical synonymic or hyponymic substitution is to be found, as shown in (c) where patas malas (bad legs) is used instead of alas quebradas (broken wings) and pico (beak) is used instead of papo (crop)/ Preliminary results demonstrate that proverbs are not frozen or immutable multi-words units kept in dusty boxes in old museums but that they take a new life when used by speakers in everyday situations. Further research that would use the same methodological approach could serve the purpose of updating, or at least completing, the data found in dictionaries and proverb collections.

Introduction

The genesis of this research is a list of fifty orally transmitted proverbs within the same family in Andalusia, southern Spain (see appendix). Most of them are related to life conditions in a rural environment during Franco's dictatorship. At that time, censorship was imposed to artistic, literary and academic activities but it also reached people's everyday life. Some issues could not be discussed, at least in public. In this context, proverbs were widely used to provide "survival strategies" to the younger generation. This research work originated to pay homage to the Celis Villalba family for their paremiological inheritance and life lessons.

Sources of the research, selection of proverbs and method of analysis

The original list of fifty orally transmitted proverbs was randomly gathered out of memory. Some of them were commonly used but others were not or seemed to have some specific features or variations. For this piece of research, we used several sources. First, the corpora of the Royal Academy of the Spanish language, that is the Corpus diacrónico del español (CORDE), a diachronic corpus that comprises texts from the early times of the Spanish language up to 1975, the Corpus del español actual (CREA), a corpus of the last quarter of the 20 th century, and the Corpus del español del siglo XXI (CORPES XXI), a corpus of contemporary Spanish of the 21 st century. Then the orally transmitted proverbs and their potential canonical forms were checked in the proverb collection by Hernán Núñez published in 1555. After consulting the corpora of the Royal Spanish Academy, we drew up a shorter list (actually seven proverbs) with specific lexico-grammatical features, including different shades of meaning between the canonical form and the orally transmitted form of the proverbs in the original list. The final step was to do research on written sources where those proverbs appeared, mainly literary works and the press.

We completed our research by searching for the canonical forms in the database of the Refranero multilingüe (multilingual proverb database) from the Centro Virtual Cervantes from which their figurative sense was taken. The Refranero multilingüe (RM) was used as a source to start classifying the proverbs according to different criteria trying to focus on what we considered specific features of the orally transmitted proverbs in the original list. We realized that there were lexical and syntactic differences between proverbs in this list and their corresponding forms in the RM and even found that some of the orally transmitted forms provided an extension functioning as a rhymed illustration of the canonical form.

Depending on the information retrieved from these previous sources and with the aim of finding earlier examples we used the archives of the Hemeroteca digital (Digital Newspaper Library) of the National Library of Spain, Google Scholar, Google Books and Europresse, a database which gives access to thousands of newspapers and magazines in several languages, thus making it possible to compile specific journalistic corpora for a given language and a given period.

For each of the seven proverbs that will be discussed below we provide the literal translation into English, the context of use of the orally transmitted form, the figurative sense of the canonical form, an analysis of the linguistic changes observed and some recorded written examples from corpora. It should be pointed out that the forms that have been chosen for the titles of each of the following sections are the orally transmitted forms and not the canonical forms.

No es señora quien lo es, sino quien lo sabe ser.

The literal translation of this proverb is 'A lady is not she who is a lady but she who knows how to be one.' This orally transmitted form is generally used by women who think very high of themselves when referring to other women who, in their opinion, do not do so-called 'traditional women's chores'. The canonical form conveys a slightly different figurative sense that can be glossed as 'What makes people respectable is not their birth but the way they conduct themselves'.

The most outstanding difference between the orally transmitted form (henceforth OTF) and the canonical form (henceforth CF) is the semantic restriction of the OTF as it refers specifically to women and how they relate to each other.

All the examples found in corpora and other written records are in the masculine, that is they use señor instead of señora. The canonical form translates as 'A gentleman is not he who is born a gentleman but he who knows how to be one': Non es señor el que señor nace, sino el que lo sabe ser (Martín Caro y Cejudo, 1792) and No es señor quien señor nace sino quien sabe serlo [START_REF] Aguilera | Refranero panameño: contribución a la paremiología hispanoamericana[END_REF] 1 . There is also a variant form of the nominal constituent (caballero instead of señor, both of them meaning 'gentleman'), found in an 18 th -century proverb collection [START_REF] Sbarbi | El refranero general español: parte recopilado y parte compuesto[END_REF].

Only one example has been found in the corpora of the Royal Spanish Academy (1) and very few in the archives of the Hemeroteca digital, exactly five, of which three include the variant form noble (nobleman) [START_REF]No es señora quien lo es sino quien lo sabe ser[END_REF]. Example ( 1) is an extract of a comedy that illustrates the use of the canonical form in literature and which, following the CF, includes nacer (be born) and señor (gentleman) whereas in the OTF the verb is ser (to be) and the noun is señora (lady).

No se puede luchar contra el destino: el que nace lechón muere cochino.

The literal sense of this proverb is 'One cannot fight against destiny, he who is born a piglet dies a pig.' This OTF is used to express a reproof towards someone who recurrently behaves or acts in the same way. The canonical form in the RM, Quien nace lechón muere cochino, is registered with the following figurative sense: 'It is impossible to change one's nature when one has received it by genetic inheritance' 2 and does not include the rhymed introduction used in the OTF, No se puede luchar contra el destino (one cannot fight against destiny), which functions as a general maxim preceding the CF to provide a down-to-earth illustration of the proverb itself.

Only one example has been found in the corpora of the Royal Spanish Academy (3), a very recent use in literature, most interestingly with a positive connotation, whereas the idea of reproof prevails in all the other examples. Example (4) has been retrieved from Google Books and includes a variant form of the rhymed introduction of the OTF, es luchar contra el sino, the first line of a quatrain with a rhyme scheme ABBA. In the case of (5), found in the archives of the Hemeroteca Digital, it should be noted that the proverb is not only used in direct speech (signaled by the quotation marks) but is also preceded by como dirían en mi tierra (as they would say in my region), which reinforces the idea of its current used in the spoken language. This proverb follows the same paremiological pattern as other proverbs with the same meaning: quien negro nace, negro muere (he who is born a black man dies a black man); quien tronco nacio, tronco murió (he who was born physically disabled, dies physically disabled), quien asno nace, asno muere (o yace) (he who is born an ass, dies an ass); quien tuerto nace, tuerto se yace (he who is born a one-eyed man, dies a one-eyed man) (Lida 1958: 27). This pattern is very productive in Spanish and particularly in the different national varieties of the Spanish language as can be observed in the quotation below: Spanish zoonymic variants meaning "fate is unavoidable" are quoted by Arora (1998), like sp. el que nació para caballo ha de morir pastando *he who was born a horse, will die grazing; el que nace tatú muere cavando [Bol.] *he who was born a tatú (armadillo), he dies digging (Ibid.); quien nace lechón muere cochino *he who was born a sucker-pig dies as a swine (Mex.); […] (Pamies-Bertrán, 2012: 295)

A veces se besan manos que se quisieran ver cortás.

This OTF is used as a survival strategy, as a piece of advice given to someone in a lower rank when dealing with a superior. It translates literally as 'Sometimes one kisses hands that one would like to see cut off.' The figurative sense of the CF, Muchos besan manos que querrían ver cortadas (Hernán Núñez, 1555) or Muchos besan manos ke kerrían ver kemadas; o kortadas [START_REF] Correas | Vocabulario de refranes y frases proverbiales[END_REF]) can be glossed as 'Sometimes one has to make an effort to be dutiful and respectful to whom one despises.'

The most relevant feature of the OTF is the inclusion of the frequency adverb a veces (sometimes) that appears in initial position in place of the pronoun muchos (many people). This leads us to infer that the advice provided should not always be followed, but only when needs be. The use of the shortened form cortás, a dialectal truncated form of the standard past participle cortadas, is the reflection of a typical oral feature in the speech of Andalusian Spanish.

Contrary to the two previous proverbs, which only had one token in corpora, there are thirteen tokens of this proverb in CORDE although none in CREA or CORPES XXI, which accounts for the relative extent of use in the past and its decay in present-day Spanish.

Most of the examples are from the 16 th century and show different types of variations, however, example (6) is the only case in which the frequency adverb a veces is used, although not in initial position. As for example [START_REF]Tiene el mal del milano, las patas malas y el pico sano[END_REF], a later use of the proverb, it should be noted the use of the variant form quemadas (burnt) instead of cortadas (cut off) and the inclusion of the interpolated metalinguistic comment decía mi abuela (my grandmother used to say). ( 6) y que las mugeres puedan / tanto, que nos humillemos / a ellas y las amemos; / pero no por esso quedan / desculpadas, / antes muy más condenadas / con sus pliegues y doblezes; / manos se besan a vezes / que devrian ser cortadas (CORDE, Cristóbal de Castillejo, 1544, Diálogo de Mujeres) [START_REF]Tiene el mal del milano, las patas malas y el pico sano[END_REF] Es como lo dice Vuesa Señoría, contestó con desfachatez el imprudente barbero; y no ha sido el menor de mis sacrificios el aparecer favorito y paniaguado del Conde de la Gomera y de algunos de sus amigos. Manos besamos que quisiéramos ver quemadas, decía mi abuela, señor Visitador. (CORDE, José Milla y Vidaurre, 1867, El visitador)

Tiene el mal del milano, las patas malas y el pico sano.

This proverb, which has a second part with a parallel opposite symmetrical pattern, translates as 'He/she is like the sick kite, bad legs and a healthy beak.' This OTF is generally used as a reproof to someone who willingly accepts an invitation to eat but rejects a suggestion to work.

The CF, El mal del milano, las alas quebradas y el pico sano, has two distinct senses. On the one hand, it refers to people who being cowardly boast about their bravery (Diccionario de Autoridades). On the other hand, it can also refer to people who being healthy and with a good appetite, pretend to be sick (Suñé Benages 2008: 115).

The OTF features lexical variation if compared with the forms found in Marqués de Santillana 1454, Hernán Núñez 1555 and Correas 1627, all of which include papo (crop) instead of pico (beak). Although the use of papo was coeval with pico in the 16 th century, as observed in [START_REF]Cuando marzo mayea, mayo marcea[END_REF] and ( 9), the noun papo was more widely used than pico at that time.

Ten tokens of the proverb have been found in CORDE although none in CREA or CORPES XXI. It is important to point out that most of the cases that appear in the different corpora go back to the 16 th century and that there is no real synchronic canonical form one can refer to, hence the importance of the OTF under analysis. This is one of the proverbs found in the Diálogo de la lengua by Juan de Valdés (González Martín, 1997: 285), El mal del milano, el ala quebrada y el papo sano, at a time, the 16 th century, when proverbs were used as illustrations to teach specific grammatical features, in this case the use of the masculine determiner el and not the feminine la before a feminine noun, ala, for euphonic reasons.

Hay ojitos que se enamoran de legañas.

The literal translation of this proverb is 'There are little eyes that fall in love with eye mucus'. This causal contradictive proverb (Dundes 1981: 57) is used to pass a personal judgment on the members of a couple in which one is more virtuous than the other, to the eyes of the speaker. This proverb is not registered in the RM, so there is no synchronic canonical form but a diachronic one Ojos hay que de lagaña se pagan 3 (Hernán Núñez 1555, Pedro Vallés 1549) whose figurative sense can be glossed as 'Taste is not always governed by reason'.

Several lexico-grammatical changes can be observed if the OTF is compared with the CF.

First of all, the hyperbaton found both in the topic and in the comment of the CF (ojos hay instead of hay ojos and de lagaña se pagan instead of se pagan de lagaña) is absent in the OTF which, besides, uses the modern verb enamorarse instead of the dated pagarse and introduces the diminutive ojitos. This is the proverb that has the greatest number of tokens in corpora, 14 tokens in CORDE, but, as is practically the general case, none in CREA or in CORPES. The relative high frequency can probably be due to the fact that the first recorded written record is found in one of the most influential literary works in Spain together with Don Quixote, that is The Tragicomedy of Calisto and Melibea, known as La Celestina (10). The proverb is then used more than a century later, this time with the verb enamorarse, as in [START_REF]Hasta el 40 de mayo no te quites el sayo[END_REF] and [START_REF]A caballo regalado no hay que mirarle el diente[END_REF] and not with the former vern agradarse.

Por una vez que maté un gato, me llamaron matagatos.

This OTF, which translates as 'Once I killed a cat and, from then on, I was called the cat-killer', is a quite transparent proverb about someone's bad reputation. It should be noted that the verb in this proverb is in the first person singular and that is used by the speaker to justify an exceptional wrong behaviour or action when they are accused of doing it regularly.

The figurative sense of the CF, Por un perro que maté, mataperros me llamaron, can be glossed as 'Sometimes an isolated case is enough to give somebody a bad reputation.' (RM).

The main difference between the CF and the OTF is lexical: perro (dog) becomes gato (cat) and mataperros (dog-killer) becomes matagatos (cat-killer). There are, however, no notable differences from a semantic point of view.

No tokens of the OTF have been found in any of the three corpora of the Spanish Royal Academy. However, several examples of the CF, such as ( 13) and ( 14), have been retrieved from the Hemeroteca digital. Event though the use of perro is the general rule, one case of the use of gato has been found [START_REF]El ojo del amo engorda el caballo[END_REF]. 'Little olive eaten, little stone away' is the literal translation of this proverb, or otherwise stated, when one has finished eating an olive, one throws the stone away, which, obviously, seems the logical thing to do. The proverb suggests that once a question or a topic is over one should stop thinking or talking about it, especially if it has been problematic or painful and recommends a course of action by advising someone to move ahead in life putting difficulties aside. The OTF uses the diminutive of the two nominal constituents of the proverb, which means that aceituna (olive) becomes aceitunita (little olive) and hueso (stone) becomes huesecito (little stone).

As can be seen in [START_REF]No se puede luchar contra el destino, el que nace lechón muere cochino[END_REF], an example from Peru and the only one found in the corpora of the Royal Spanish Academy, it is the CF, without the diminutives, that is used. [START_REF]No se puede luchar contra el destino, el que nace lechón muere cochino[END_REF] El verdugo, comprendiendo la burla de estas palabras, le contestó: -Holgárame de no ser pagado, que la paga ha de ser tal, que después que concluya con estos compañeros, venga yo a hacer cabal la docena del fraile. Aceituna comida, hueso fuera. (CORDE, Ricardo Palma, 1877, Tradiciones peruanas, cuarta serie)

All the other examples, ( 16), ( 17) and ( 18), were found in the Hemeroteca digital, include one or two diminutive forms and come from the press form South-western Spain. As with (5), example ( 18) also shows the importance of using the press as a corpus since, in these two instances, either proverb is used in direct speech, which demonstrates the use of the proverbs in the contemporary spoken language. [START_REF][END_REF] No saben ná los alemanes... Han conseguido lo que siempre buscaron: tener a sus pies a toda la mano de obra que necesiten a precio de saldo. Deben su bienestar a que los demás países le compran sus medicamentos, coches, trenes, tanques y demás, pero "aceitunita comida, huesecito fuera". Two other pieces of evidence of the Andalusian origin of this OTF are its inclusion in Modismos comparativos de Andalucía (Comparative Idioms from Andalusia), in which the proverb appears as Aceituna comida, huesecito fuera (Antequera Luengo, 2007: 17), but also the fact that Aceituna comida, hueso fuera is one of the many proverbs collected from informers by the Spanish writer Fernán Caballero 4 when living in Andalusia (Jarilla Bravo, 2019: 178).

Conclusions

Orally transmitted proverbs should be used as a benchmark against canonical forms and other written forms as found in texts since a thorough description of a given paremiological unit should include all its variant forms and detailed information about nonstandard semantics and pragmatics.

The search for examples in corpora shows that some of them are either non-existent (Por una vez que maté un gato me llamaron matagatos) or nonce forms (No es señor quien señor nace sino quien sabe serlo, Quien nace lechón muere cochino, Aceituna comida, hueso fuera).

However, the daily press provides examples of actual uses of proverbs that account for the vitality of their variant forms both in spoken language (direct or indirect speech) and in specific regional contexts, which is extremely helpful for acknowledging orally transmitted forms not as nonce formations or deviations but as real variant forms.

Collecting the greatest number of examples is essential for research on the evolution of paremiological semantics. The diversity of contexts of use will allow the researcher to identify and check whether the semantic specificities of the orally transmitted form are reflected or not in recorded written sources.

Endnotes

1.

Even though two references are provided, in fact they can be subsumed to only one canonical form as the difference between the two (el que / quien) are exact grammatical equivalents (he who / who).

2.

The analogical equivalent proverb in English is "A leopard can't change its spots" (used to say that people cannot change their basic personalities, habits, etc.).

3.

The verb pagarse in the sixteenth century is used in the same sense as the verb enamorarse (to fall in love) in contemporary Spanish.

4.

Fernán Caballero was the pseudonym of Cecilia Böhl de Faber y Ruiz de Larrea, her real name.

( 1 )( 2 )

 12 PRÍNCIPE Id con Dios. / MÚSICO 1 El cielo dé a vuestra alteza, señor, un siglo de vida, amén. / PRÍNCIPE Id con Dios. / MÚSICO 1 ¡Qué gran valor! / MÚSICO 2 ¡Qué cordura! / MÚSICO 1 Octavio, ven. No es señor quien señor nace, sino quien lo sabe ser. (CORDE, Luis Vélez de Guevara, c. 1630, Reinar después de morir) Bienaventuradas las provincias que tienen autoridades de iniciativas propias, y bienaventurados los pueblos que saben cumplir una de las Obras de Misericordia, que no es noble quien noble nace, sino quien lo sabe ser; (El Globo, Madrid, 1875)

( 3 )

 3 No conocía otra vida. Nació en un palacio rodeado de comodidades y sirvientes y murió en otro sin que le faltara bienestar y asistentes. Porque el que nace lechón, muere cochino. (CORPES XXI, Nieves Concostrina, 2018, Pretérito imperfecto) (4) Es luchar contra el sino / Con que vienen al mundo ciertas gentes, / Querer hacerlas pulcras y decentes: / El que nace lechón muere cochino. ("El lavatorio del cerdo", 1861-62, Miguel Agustín Príncipe y Vidaud, Fábulas en verso castellano) (5) Para el portavoz adjunto del Grupo Parlamentario Popular, Luis Ramallo, "cada uno se retrata como es, y como dirían en mi tierra, el que nace lechón muere cochino". (El País, 12/04/1988)

( 8 )( 9 )

 89 Elicia. Matado sea tal garguero. Por ella se puede dezir, el mal del milano, las alas quebradas y el papo sano / Areusa. Y cómo se dirá con razón esso y más, pues con tal herida tiene tales alientos.(CORDE, Gaspar Gómez de Toledo, 1536, Tercera parte de la tragicomedia de Celestina) Dixo don Francisco Fenollet: Ximeno, por su mal conoce el ajeno. / Dixo Joan Fernandez: Don Francisco, vos no quereis acabar de conoscer ese milan; por él se dixo: El mal de milano, las alas quebradas y el pico sano.(CORDE, Luis Milán, 1561, El Cortesano)

( 10 )

 10 ¿A quién "gentil"? ¡Mal me haga Dios si ella lo es ni tiene parte dello, sino que hay ojos que de lagaña se agradan! Santiguarme quiero de tu necedad y poco conocimiento. (CORDE, Fernando de Rojas, 1499, La Celestina) (11) Rojas.-Ésa misma, pues tenía muy mala cara, era un poquita sucia y no sé si tuerta; y sobre todo, más vieja que el alcabala. / Sol.-Señor, ojos hay que de lagaña se enamoran. / Rojas.-Quien feo ama, hermoso le parece. (CORDE, Agustín de Rojas Villadrando, 1603, El viaje entretenido) (12) Vase Valenzuela y sale Gálvez. / Gálvez. ¡Que haya mujer de tan mal gusto! Por ésta se debió de decir que hay ojos que de legañas se enamoran. (CORDE, Anónimo, c. 1611, El hospital de los podridos)

( 13 )

 13 Pero esta ocurrencia no ha de ser causa de lo que vulgarmente se dice de "por una vez que maté un gato me pusieron mata-gatos" (El Eco del Comercio, Madrid, 22/05/1846) (14) -Si, si… buenas felpas me costaron tus torpezas. Parecia que de intento tropezabas siempre con rivales ó maridos… -Qué! señor, por una vez que maté un perro… Ya no os acordais de mis buenos servicios. (El Corresponsal, Madrid, 22/05/1843) Aceitunita comida, huesecito fuera.

(

  Diario de Huelva, 29/03/2014) (17) Antañazo era lo de aceitunita comida huesecito fuera, esto es lo que admite el estadio propio y de ahí no se puede pasar. Es más, entonces ni a soñar que la demanda fuese a acercarse a la oferta. (Diario de Sevilla, 13/07/2022) (18) "Aceitunita comida, huesecito tirado". Son las palabras de Joaquín Caparrós cuando se le pregunta por la planificación de un Sevilla que se encuentra ya en plena competición oficial. (El País, Sevilla, 03/08/2018)