J Jarry

J L Faucheron

W Moreno

C A Bellera

S Evrard
email: evrard@bergonie.org

Prof S Evrard

Is delayed colo-anal anastomosis an alternative to prophylactic diverting stoma after total mesorectal excision for middle and low rectal carcinomas?

Keywords: diverting stoma, rectal cancer, delayed colo-anal anastomosis, fistula, proctectomy, total mesorectal excision

Background: After total mesorectal excision (TME), a low colorectal or colo-anal anastomosis is usually performed. A prophylactic covering stoma is often used, especially for patients receiving neoadjuvant chemo-radiotherapy. However, morbidity is high, mainly due to anastomotic leakage.

Methods: From May 2000 to October 2008, patients with middle or low rectal cancer who underwent a trans-anal pull-through procedure after TME were prospectively recorded. No covering stoma was performed in these patients. However, they all underwent a delayed colo-anal anastomosis (DCA), which was performed 6 days following the TME, on average. Both the surgical technique and follow-up were standardized. Patients with T3, T4 and/or N+ cancers were given pre-operative radiotherapy. A retrospective analysis was done to assess post-operative mortality, morbidity, and oncologic and functional results.

Results:

One hundred consecutive patients with rectal tumours at a median distance of 5 cm from the anal verge underwent DCA after TME. The 5-year overall and disease-free survival rates were 81% and 66%, respectively. The post-operative mortality rate was 3% and the overall post-operative morbidity rate was 36%, with only 3 anastomotic leakages.

After two years, 73% of the patients had good functional outcomes.

Conclusion:

The trans-anal pull-through procedure after TME, followed by DCA seems to be a safe and efficient sphincter-preserving procedure to treat patients with middle or low rectal cancer while avoiding a prophylactic, diverting stoma.

Introduction

The surgical treatment of lower rectal cancers has evolved from abdomino-perineal resection to proctectomy with TME and colo-anal anastomosis. The main drawback of colo-anal anastomosis is the risk of leakage, which is reported to occur in 2.9% -20% of cases [START_REF] Vignali | Factors associated with the occurrence of leaks in stapled rectal anastomosis: a review of 1014 patients[END_REF][START_REF] Hautefeuille | Functional and oncologic results after colo-anal anastomosis[END_REF]. In half of these cases, the anastomotic leakage requires an additional operation [START_REF] Nesbakken | Outcome and late functional results after anastomosis leakage following mesorectal excision for rectal cancer[END_REF], and a third of these require the creation of a definitive colostomy [START_REF] Eriksen | Anastomotic leakage following routine mesorectal excision for rectal cancer in a national cohort of patients[END_REF]. Furthermore, anastomotic leakage represents the third main cause of post-operative fatalities in rectal surgery after myocardial infarction and bronchopneumonia [START_REF] Longo | Outcome after proctectomy for rectal cancer in Department of Veterans Affairs Hospitals: a report from the National Surgical Quality Improvement Program[END_REF]. It also results in a higher prevalence of local, tumoral, recurrence [START_REF] Bell | Anastomotic leakage after curative anterior resection results in a higher prevalence of local recurrence[END_REF]. In order to reduce the burden of the anastomotic leakage, a prophylactic covering stoma is frequently created. However, this stoma can result in minor to severe complications in 10% of cases [START_REF] Edwards | Stoma-related complications are more frequent after transverse colostomy than loop ileostomy: a prospective randomized clinical trial[END_REF][START_REF] Law | Randomized clinical trial comparing loop ileostomy and loop transverse colostomy for faecal diversion following total mesorectal excision[END_REF]. Additionally, the construction of a stoma is generally regarded as an unfavourable outcome, since the quality of life experienced by stoma patients is considered inferior to that of non-stoma patients [START_REF] Hida | Functional outcomes after low anterior resection with low anastomosis for rectal cancer using the colonic J-pouch. Prospective randomized study for determination of optimum pouch size[END_REF]. Finally, it imposes significant medical costs for society. In 1932, Babcock described a new technique of trans-anal colonic pull-through [START_REF] Babcock | Operative treatment of carcinoma of the rectosigmoid[END_REF]. More recently, Baulieux [START_REF] Baulieux | Oncologic and functional results of resections with direct delayed colo-anal anastomosis in previously irradiated cancers of the lower rectum[END_REF] proposed an operative technique halfway between Babcock's pull-through and Park's direct colo-anal anastomosis [START_REF] Parks | Resection and sutured colo-anal anastomosis for rectal carcinoma[END_REF] that is now performed routinely in our hospitals. In comparison to direct colo-anal anastomosis, this new surgical concept has a main theoretical advantage: a prophylactic stoma does not need to be performed due to the very low risk of anastomotic leakage. In this retrospective study we present the results of 100 consecutive delayed colo-anal anastomosis (DCA).

Methods

Patient selection

The trans-anal pull-through procedure followed by a DCA was the standardised operation for patients with middle and low rectal cancer in two French university medical centres (Grenoble Hospital and the Regional Cancer Centre Bergonié in Bordeaux). A patient database was compiled prospectively within the two institutions. The preoperative assessment included a digital rectal examination, a colonoscopy with biopsy, a CT scan of the thorax, abdomen and pelvis and a pelvic MRI. The precise level of the lower edge of the tumour from the anal verge was assessed by the surgeon. Patients were classified according to the UICC classification. Patients with T3, T4 and/or N+ disease received preoperative radiotherapy or chemo-radiotherapy. Surgery was performed 6 weeks on average after radiotherapy. We performed a preoperative evaluation of anal sphincter function for all patients by digital anal exploration. No patient suffered from faecal incontinence or had a hypotonic sphincter -both of which would have necessitated an abdominoperineal resection. The preoperative anaesthetic evaluation was done using the American Society of Anesthesiology guidelines. The criteria for laparoscopic approach was the presence of a tumour limited to the rectum and the mesorectum, without any extension to adjacent organs, and regardless of the distance of the lesion from the anal verge.

Operative procedure

The surgical procedure of laparoscopic TME followed by colonic pull-through and DCA involved two stages. In the first stage, the patient was placed in a Lloyd-Davies position and an abdominoperineal approach was used. This stage was performed laparoscopically, unless there were criteria for open surgery. The surgical procedure began with the division of the inferior mesenteric vessels. The splenic flexure was then systematically detached and the left colon completely mobilized. A laparoscopic TME was performed until the pelvic floor was reached.

We then began a perineal phase. An anal retractor was used to facilitate the exposure of the anal canal. After infiltrating the sub-mucosal plane of the upper anal canal with a saline and adrenaline solution, a circumferential incision of the mucosa was performed at the level of the dentate line. This was followed by a short mucosectomy. The rectum was then dissected along the perineal plane until the level of the abdominal dissection was reached.

The rectum and sigmoid colon were then pulled through the anal canal and cut at the level of the ligation of the inferior mesenteric artery. A colonic segment of about 10 cm was left outside the anal canal and tied by two stitches to the right buttock (Figures 1 and2). After this perineal phase, we returned to the laparoscopic phase to insert a pelvic suction drain.

Between the two stages, the vascularisation of the colonic segment was checked daily in order to detect any colonic necrosis.

The second surgical stage of operation was performed around the sixth post-operative day, under general anaesthesia with curarisation to relax the pelvic floor. The patient was placed in the lithotomy. No retractors were needed, and the adhesions between the anal canal and colon had to be conserved. After tying off the mesocolon at the level of the anal verge, the colonic pull-through segment was cut and a hand sewn, colo-anal anastomosis was performed using interrupted sutures at the dentate line level. In all 100 consecutive procedures performed, no covering stoma was necessary.

Follow-up

A standardized follow-up was completed, one month post-operatively, then every four months during the first two years, and every six months in the third through the fifth year.

The follow-ups included clinical examination, CEA level determination, and CT scans of the thorax and abdomen. Patients with confirmed lymph node metastases received postoperative chemotherapy. Functional results over time were evaluated using questionnaires, independently completed by the patients during each follow-up consultation.

Endpoint assessment

The endpoints of the study were the oncologic results, post-operative morbidity and mortality, and functional outcomes. The oncologic results included the rate of microscopic resection, local recurrence, distant metastases and survival rates. Complete microscopic resection was defined as R0 resection and included both distal and circumferential negative margins. Overall survival (OS) is defined as the delay between the date of surgery and date of death. Disease-free survival (DFS) is defined as the delay between the date of surgery and the date of the first post-operative diagnosis of local recurrence, distant metastases, or death. Initial post-operative morbidity results were accounted for up to 30 days postoperatively. Morbidity cases were recorded according to the classification of Dindo [START_REF] Dindo | Classification of surgical complications. A new proposal with evaluation in a cohort of 6336 patients and results of a survey[END_REF].

Functional results of the patients were evaluated using the Wexner continence score [START_REF] Jorge | Etiology and management of fecal incontinence[END_REF],

and the Hida scoring system [START_REF] Hida | Functional outcomes after low anterior resection with low anastomosis for rectal cancer using the colonic J-pouch. Prospective randomized study for determination of optimum pouch size[END_REF]. Wexner scores between 0 and 10 were defined as "very good" or "good", while scores greater than 10 were defined as "poor" or "very poor".

Statistical methods

Survival endpoints, including OS and DFS, were estimated using the Kaplan-Meier method. Quantitative continuous variables were presented using descriptive statistics including the median and extreme values. Qualitative variables were described using counts and proportions. All analyses were carried out using the SAS version 9.2 software for Windows (SAS Institute Inc., Cary, NC, USA.).

Results

Population study (Table 1)

From May 2000 to October 2008, 100 consecutive patients underwent a pull-through procedure after TME, followed by a DCA. The patient group was comprised of 70 men and 30 women with a median age of 64 years (range . The median distance between the inferior margin of the tumour and the anal verge was 5 cm (range 2-12). Using the American Society of Anaesthesiology classification system, 48 patients were classified as ASA 1, 34 as ASA 2, and 18 as ASA 3. Using the post-operative UICC staging, 16 patients were classified as stage 0 (complete histological response), 20 as stage I, 23 as stage II, 26 as stage III, 10 as stage IV and 5 could not be classified (missing data). Seventy patients received preoperative radiotherapy (the median dose applied to the pelvis was 45Gy), 18 of whom received concomitant chemotherapy (5-fluorouracil). Thirty patients did not receive neoadjuvant treatment.

Operative data

Seventy five patients underwent laparoscopy and twenty-five patients underwent open surgery. Among the laparoscopic cases, seven conversions were performed. The median time for the first operative stage was 276 minutes (range 150-450) and 21 minutes (range for the second stage. The median interval between first and second stages was six days (range 2-10) and the median hospital stay was fifteen days (range 7-157).

Oncologic results

Resection was classified as R0 in 92 cases and R1 in 8 cases. None of the resections were classified as R2. Assessment of recurrence and survival was carried out with a median follow-up of 38.4 months (range 28.95-45.54). During the follow-up period, 20 patients presented a distant metastasis with a median delay of 10 months (range 1-24). Seven patients developed a local recurrence at a median time of 16 months (range 8-29) following surgery. Fourteen patients died during the follow-up period, of which ten deaths were attributed to the disease and four to other causes. The disease-free survival rate was 84% at 1 year and 66% at 5 years and the overall survival was 94% at 1 year and 81% at 5 years (Figure 3).

Postoperative morbidity and mortality

The postoperative mortality rate was 3%. This included two deaths due to cardiac complications on the 12 th and 25 th post-operative day and one further death on the 25 th postoperative day due to brain metastasis for patients who were asymptomatic before the surgery. A total of 40 post-operative complications (22 surgical and 18 medical) occurred in 36 patients, representing an overall early morbidity rate of 36%. According to the classification of Dindo et al. [START_REF] Dindo | Classification of surgical complications. A new proposal with evaluation in a cohort of 6336 patients and results of a survey[END_REF], there were 8 grade I, 11 grade II, 8 grade IIIa, 13 grade IIIb, 0 grade IV and 3 grade V complications. Among the surgical complications (Table 2), ten patients were diagnosed with pelvic sepsis complications (six pelvic abscesses without evident fistula, one infected pelvic haematoma, two anastomotic fistulas and one rectovaginal fistula). Fourteen patients required an additional operation including three Hartmann's procedures, one temporary colostomy and three temporary ileostomies. Two patients had a necrosis of the exteriorised colonic segment after the first stage of the procedure. Late post-operative complications appeared in 18 patients, which consisted of 15 anastomotic stenoses, 2 colonic prolapses and 1 rectovaginal fistula requiring a temporary colostomy.

Functional outcomes

Of the 100 patients, 6 had no functional evaluation, either because they were not present for the follow-up or because they had a stoma at the time of the evaluation. Two patients required a definitive diverting stoma for major anal incontinence. The median Wexner score was 10 during the first post-operative year, but improved to 9 at the second postoperative year and 7.8 beyond the second post-operative year. During the first year, only 40 patients had good or fair functional results. Beyond the second post-operative year, functional results improved rapidly in both groups with 73% of results being good or fair (Figure 4). Analysis of the Hida scoring system reveals that during the first post-operative year, 20 patients had more than 6 stools per day while another 20 had less than 3 stools per day. After one year, only 7 had more than 6 stools per day and 50 patients had less than 3 stools per day. One third of all patients had no stools at night during the first post-operative year, which increased to over half in the second post-operative year. Additionally, 54 patients were able to wait more than 30 minutes before using the toilet during the first postoperative year, and the number of patients in this category rose to 70 after the first year. An overwhelming majority (n=94) of the patients used a pad during the first post-operative year. However, 50 of these patients stated that the pads were used simply as a precautionary method. This number dropped slightly to 73 after the first post-operative year. Additionally, after one year, 75 patients had no usual stool incontinence, all reported perfect continence for gas, 73 could distinguish stools from gas, and 75 had sufficient continence to carry out a normal social life. Overall, 51 patients accepted treatment to improve bowel function and 35 followed a solid food diet after one year.

Discussion

Does delayed colo-anal anastomosis prevent anastomotic leakage after TME? If so, is it the next step up in rectal surgery?

Principles of DCA

Since 1950, only a few surveys about DCA have been published [START_REF] Charrier | Forty cases of rectal cancer treated by the Babcock method[END_REF][START_REF] Bacon | Present status of the pull-through sphincter-preserving procedure[END_REF][START_REF] Maliakas | Babcock-Bacon type abdominotransanal amputation. Indications and results apropos of 50 cases[END_REF][START_REF] Kirwan | Pull-through operations with delayed anastomosis for rectal cancer[END_REF][START_REF] Khubchandani | The Bacon pull-through procedure[END_REF][START_REF] Olagne | Functional results of delayed colo-anal anastomosis after preoperative radiotherapy for lower third rectal cancer[END_REF][START_REF] Remzi | Outcomes following turnbull-Cutait abdominoperineal pull-through compared with colo-anal anastomosis[END_REF]. With fistula rates ranging from 0 to 7% and pelvic abscess rates from 2 to 7%, local septic complication rates with DCA appear lower in comparison to direct colo-anal anastomosis (Table 3). Three hypotheses could explain these results. Firstly, at the end of the first surgical stage, the pulled-through colon is free from attachment in the anal sphincter. Consequently, the pelvic floor rises up after the effects of the curarisation disappear, and there is no anastomotic tension (as opposed to a direct colonal anastamosis). Secondly, before the DCA is performed, the pulled-through colon and the anal canal have already started to circumferentially adhere together. This adhesion limits the risk of potential fistulae.

Thirdly, the risk of anastomotic leakage due to early necrosis of the descending colon by vascular occlusion of the Riolan's arcade is theoretically lower. This is largely due to the fact that the colonic pulled-through segment is inspected daily, therefore resulting in a quicker diagnosis of necrosis. Thus, the patient can be re-operated immediately before the 6th post-operative day, creating an opportunity for the surgeon to remove the necrotic colon and to perform a new procedure by pulling the transverse colon through the anal canal followed by a DCA. By doing so, the patient avoids a Hartmann's procedure. It is important not to confuse between necrosis of the descending colon and necrosis limited to the exterior segment only. Differential diagnosis is made by rectoscopy. Although rare (only two cases in our series), necrosis of the exterior colon may appear in patients with tight sphincters. However this is of little consequence since it will be removed during the second stage of the procedure.

Theoretical advantages

Although a DCA could be done either by open surgery or via laparoscopy, we prefer to employ the latter since it offers a better vision of the pelvis. This allows us to better visualize the hypogastric and pelvic plexuses which must be avoided during the operation.

Laparoscopy also reduces the risk of developing long-term incisional hernias, which are associated with open surgery [START_REF] Laurent | Long-term wound advantages of the laparoscopic approach in rectal cancer[END_REF]. Furthermore, we do not perform an abdominal incision to remove the colo-rectal segment, which is instead pulled through the anal canal [START_REF] Faucheron | Laparoscopic proctectomy and delayed coloanal anastomosis for lower rectal cancer: 42 consecutive cases[END_REF].

Consequently, the first stage of our operative procedure is shorter than a laparoscopic proctectomy with TME followed by direct colo-anal anastomosis since neither a covering stoma nor a colonic reservoir is needed. In comparison with open surgery, post-operative recovery is therefore quicker, less stressful since there is neither a stoma nor an abdominal incision (except the port incisions), and, due to lower pain levels, has reduced analgesia requirements. Furthermore, bowel function returns earlier, allowing patients to drink water on the first post-operative evening and to stand up on the second day in accordance with fast-track surgery procedures [START_REF] Zargar-Shoshtari | Fast-Track surgery may reduce complications following major colonic surgery[END_REF]. Although the median time of hospitalization following our procedure appears long (15 days), it is important to keep in mind that the direct coloanal anastomosis requires an additional hospitalization stay to close the stoma. These two hospitalizations generally amount to a longer overall hospital stay than DCA. From an oncological point of view, there is increasing evidence that laparoscopy provides equivalent oncologic results to open surgery for rectal cancer [START_REF] Jayne | Randomized Trial of laparoscopic-assisted resection of colorectal carcinoma: 3-year results of the UK MRC CLASSICC Trial Group[END_REF][START_REF] Laurent | Laparoscopic versus open surgery for rectal cancer: long-term oncologic results[END_REF][START_REF] Lujan | Randomized clinical trial comparing laparoscopic and open surgery in patients with rectal cancer[END_REF].

The main advantage of DCA is the avoidance of a covering stoma, thereby significantly improving the patient's quality of life. From a financial point of view, it should also reduce the costs of rectal cancer surgical management since there is no need for stapler devices, no stoma pouches, no nurses required to attend to the stoma, and no subsequent operations to close the temporary stoma. A financial comparative study between delayed and direct coloanal anastomosis after TME should be done to confirm this hypothesis.

Theoretical functional drawbacks

The main criticism of the DCA procedure surrounds the poor functional outcomes due to the absence of a colonic reservoir. A frequent result of direct colo-anal anastomosis is fractionation of the patient's stools as well as an increase in the degree of urgency.

Consequently, some surgeons opt for a J-shaped colonic reservoir, which typically has better short-term functional outcomes [START_REF] Hallböök | Randomized comparison of straight and colonic J pouch anastomosis after low anterior resection[END_REF][START_REF] Dennett | Misconceptions about the colonic J-pouch: what the accumulating data show[END_REF]. While the addition of a reservoir does not improve continence, it does reduce frequency and urgency of stools [START_REF] Araki | Functionnal results of colonic J pouch anastomosis for rectal cancer[END_REF][START_REF] Dehni | Results of colo-anal anastomosis for rectal cancer[END_REF][START_REF] Ho | Prospective randomized controlled study of clinical function and anorectal physiology after low anterior resection: comparison of straight and colonic J pouch anastomoses[END_REF], and this starting as soon as the first post-operative year [START_REF] Ortiz | Colo-anal anastomosis: are functional results better with a pouch?[END_REF]. Nonetheless, these functional advantages seem to diminish over time. Indeed, all studies of straight colo-anal anastomosis show improving functional outcomes after the first year, presumably because of an acquired compliance of the neorectum. Heriot et al.'s meta-analysis confirms these findings by demonstrating that at 2 years post-operatively, the difference in stools between straight and J-pouch colo-anal anastomosis is only 0.74 stools per day [START_REF] Heriot | Meta-analysis of colonic reservoirs versus straight colo-anal anastomosis after anterior resection[END_REF]. [START_REF] Heriot | Meta-analysis of colonic reservoirs versus straight colo-anal anastomosis after anterior resection[END_REF]. We observe similar results in our current study with 73% of good functional outcomes after two years. Recently, Remzi et al. reported similar results based on 44 DCA and concluded that functional outcomes were comparable between delayed and direct hand-sewn coloanal anastomosis groups [START_REF] Remzi | Outcomes following turnbull-Cutait abdominoperineal pull-through compared with colo-anal anastomosis[END_REF]. Moreover, constructing a reservoir requires a longer operative time and is not technically possible for all the patients [START_REF] Harris | Reasons for failure to construct the colonic J-pouch. What can be done to improve the size of the neorectal reservoir?[END_REF]. Additionally, the Jpouch has been shown to increase the difficulty for the neo-rectum to empty [START_REF] Berger | Excision of the rectum with colonic J pouch-anal anastomosis for adenocarcinoma of the low and mid rectum[END_REF].

Recently, Lange et al. [START_REF] Lange | Faecal and urinary incontinence after multimodality treatment of rectal cancer[END_REF] argued that post-operative incontinence after TME occurred due to intra-operative injuries to the innervation of the levator ani. Thus, this main negative functionnal result seems to be independent from construction of a reservoir.

Conclusion

Should the relatively minor improvements in functional results offered by colonic reservoirs outweigh the safety and financial benefits of DCA in the management of low and middle rectal cancer? In the most complexe cases, the Cleveland Clinic Foundation's [START_REF] Remzi | Outcomes following turnbull-Cutait abdominoperineal pull-through compared with colo-anal anastomosis[END_REF] recommends DCA as the appropriate procedure to use before creating a permanent stoma. Indeed this is a first step towards full recognition of DCA. We also advocate its use as a routine procedure in middle and low rectal cancer. In addition, our retrospective data has enabled us to construct a prospective, multicentric, phase 2 study, which is currently ongoing.

Table 1 .

 1 Demographic data and tumour characteristicsSex (

Table 2 .

 2 Surgical morbidity (according to the classification of Dindo et al.14 )

	Anastomotic fistula (IIIb)	2
	Rectovaginal fistula (IIIb)	1
	Perianastomotic abscess (IIIa)	6
	Pelvic hematoma (IIIb)	1
	Colonic ischemia (IIIb)	4
	Colonic perforation (IIIb)	2
	Occlusion (IIIb)	1
	Evisceration (IIIb)	1
	Necrosis of the exteriorised colonic segment (IIIa)	2
	Prostatic bleeding (IIIb)	1
	Perianal abscess (IIIa)	1
	Total	22

Table 3 .

 3 Morbidity in direct coloanal anastomosis studies

	Authors	Preventive diverting stoma	Anastomotic leak (%)	Pelvic abscess (%)
	Parks et al 11	100%	7.9	10.5
	Lazorthes et al 22	100%	4.6	4.6
	Hautefeuille et al 2	100%	20	__
	Benchimol et al 23	100%	8.5	__
	Berger et al 24	100%	3	1.23
	Rullier et al 25	100%	10	3
	Our series of ACAD	0%	3	6

Aknowledgments

We gratefully acknowledge Ms Antonia Giraud, Suzanne Gaskell and Pippa McKelvie for their help with the manuscript.


Guillaume Chapuy

Ines Klimann

On the supports of recognizable series over a field and a single letter alphabet

.

Introduction

A series over a semiring S and an alphabet Σ is a function S : Σ * → S which maps a word on Σ to an element of S. The series is recognizable if it is the behaviour of a weighted finite automaton. Its support is the set of all words which are not mapped to zero.

Recently Kirsten introduced the notion of SR-semirings and characterized them algebraically [START_REF] Daniel | An algebraic characterization of semirings for which the support of every recognizable series is recognizable[END_REF]. An SR-semiring is a semiring S such that the support of any recognizable series over S and any finite alphabet Σ is a recognizable language. Many semirings are SR-semirings; typical examples of non-SR-semirings are Z and Q.

Besides an algebraic characterization, Kirsten showed that in the definition of an SR-semiring, one could equivalently restrict the alphabet Σ to any fixed finite alphabet of at least two letters. More precisely, given a finite alphabet Σ of at least two letters, a semiring S is an SR-semiring if and only if for every recognizable series S : Σ * → S the support of S is a recognizable language. In [4, p.500] Kirsten asked whether one can even restrict Σ to a single letter alphabet: Given a single letter alphabet Σ and a semiring S, if the support of every recognizable series Σ * → S is a recognizable language, is S necessarily an SR-semiring?

In this note we negatively answer Kirsten's question, by showing that the support of a recognizable series over Q and a single letter alphabet is always recognizable. In fact we prove this property for series over any field of characteristic zero. We also show that this property cannot be extended to fields of characteristic p = 0: We construct a recognizable series over a field of prime characteristic and a single letter alphabet whose support is not recognizable. This provides an example of a non-SR-semiring which does not include Z, answering a question of Kirsten and Quaas [START_REF] Kirsten | Recognizability of the support of recognizable series over the semiring of the integers is undecidable[END_REF].

In Section 2 we present basic definitions on series and weighted finite automata. In Section 3 we prove that the support of a recognizable series over a field of characteristic zero and a single letter alphabet is recognizable. Finally in Section 4 we give an example of a recognizable series over a field of prime characteristic and a single letter alphabet whose support is not recognizable.

Preliminaries

Basic notions on (recognizable) languages and finite automata are supposed to be known. The reader can refer to [START_REF] Sakarovitch | Elements of Automata Theory[END_REF] for details. Remind in particular that recognizable languages over a single letter alphabet {a} are exactly the sets a N where N ⊆ N is ultimately periodic -possibly finite (this is immediate considering deterministic automata).

A monoid is a set with a binary associative operation and an identity element for this operation. A monoid whose operation is commutative is called a commutative monoid. A semiring (S, +, •, 0, 1) consists of a set S equipped with two binary operations: product • (or simply denoted by juxtaposition) and sum + such that (S, +, 0) is a commutative monoid, (S, •, 1) is a monoid and product distributes over sum. If no confusion arises we will denote it by S.

Matrices over a semiring S can be equipped with usual matrix multiplication.

A series S over a semiring S and an alphabet Σ is a map from Σ * into S. The image of a word w is its coefficient and is denoted by (S, w).

A weighted finite automaton (WFA) A over a semiring S and a single letter alphabet {a} is a tuple (Q, M, I, F ) where

• Q is a finite set of states, • M ∈ S Q×Q is a Q × Q-matrix of transitions, • I ∈ S Q is the initial vector, • F ∈ S Q is the final vector.
We say that a state is initial (resp. final) if its coefficient in I (resp. F ) is not zero, this coefficient is called the initial (resp. final) weight of the state. The automaton is normalized if it has a single initial state i with weight 1, a single final state f with weight 1 and the i-th column and the f -th line of M have only 0-inputs (speaking in graphical terms: no transitions are leading to the initial state and no transitions are leaving the final state).

The series recognized by A, also called its behaviour, is the series

|A| : {a} * → S : a n → t IM n F.
A series is recognizable if it is the behaviour of an automaton. The support Supp(S) of a series S is the language of all words which are not mapped to zero.

Note that the notions of weighted finite automata and series recognized by such automata can be defined in the same way for any finite alphabet, see [START_REF] Berstel | Noncommutative Rational Series with Applications[END_REF][START_REF] Droste | Handbook of Weighted Automata[END_REF][START_REF] Sakarovitch | Elements of Automata Theory[END_REF] for more details.

Main result in characteristic zero

This section is devoted to the proof of our main result which is given by the following proposition: Proposition 1. The support of a recognizable series over a field of characteristic zero and a single letter alphabet is recognizable.

Let A = (Q, M, I, F ) be a WFA over a field of characteristic zero S and a single letter alphabet {a}. It is well-known that there exists a normalized WFA A ′ such that (|A|, w) = (|A ′ |, w) for every non-empty word w [3, Th. 2.11]. Therefore we can suppose that A is normalized without loss of generality.

Let us denote respectively by i and f the initial state and the final state of A: the coefficient of the word a n in the behaviour of A is the (i, f )-input of the matrix

M n : (|A|, a n ) = (M n ) i,f .
We denote by (u n ) n∈N the sequence ((M n ) i,f ) n∈N .

Lemma 1. The sequence (u n ) n∈N is the solution of a linear recursion with constant coefficients.

Proof. The classical Cayley-Hamilton Theorem states that every square matrix over a commutative ring (and hence over a field) satisfies its own characteristic equation [START_REF] Michael | Introduction to Commutative Algebra[END_REF]. A direct consequence of this theorem is that any square matrix is the zero of a non-trivial polynomial equation with leading coefficient 1.

As a result, we have

M |Q| = α |Q|-1 M |Q|-1 +• • •+α 1 M +α 0 I |Q| , for some (α 0 , . . . , α |Q|-1 ) ∈ S |Q| ,
where I |Q| is the identity |Q| × |Q|-matrix and hence for all k ≥ |Q|:

M k = α |Q|-1 M k-1 + • • • + α 1 M k-|Q|+1 + α 0 M k-|Q| .
Taking the (i, f )-inputs of all these matrices we thus obtain

u k = α |Q|-1 u k-1 + • • • + α 1 u k-|Q|+1 + α 0 u k-|Q| .
We can now apply a result of Lech [START_REF] Lech | A note on recurring series[END_REF][START_REF] Mollin | Number theory and applications[END_REF]: in a field of characteristic zero, if the values of the terms of a sequence solution of a linear recursion equation with constant coefficients are infinitely often zero, then they are equal to zero ultimately periodically. Since recognizable languages over a single letter alphabet are the ultimately periodical languages, Proposition 1 is a direct consequence of Lemma 1.

Counter-example in characteristic p

Lech proved by giving an example that his result on sequences cannot be extended to fields of prime characteristic [START_REF] Lech | A note on recurring series[END_REF]. In this section, we adapt this example to prove that neither can be Proposition 1.

Remember that a field has zero or prime characteristic. Denote by F p the cyclic field of prime size p and consider F p (X) the field of fractions of the polynomial ring in one indeterminate F p [X], i.e. the smallest field in which F p [X] can be embedded: it has characteristic p.

The series S : {a} * → F p (X) given by

a n → (S, a n ) = (1 + X) n -1 -X n
is recognized by the automaton of Figure 1. We denote by L the complement of its support. We have for each n:

| 1 | 1 | -1 | 1 | -1 | 1 a | 1 + X a | 1 a | X
(S, a n ) = (1 + X) n -1 -X n = n-1 k=1 n k X k . (1)
Now, the greatest power of p dividing n k is the number of carries by substracting k to n in base p (see [START_REF] Weil | Collected papers -Ernst Eduard Kummer, volume 1 (Contributions to number theory[END_REF]). Therefore we have:

• If n is a power of p, say p α , it is written 10 α in base p and any k (1 ≤ k ≤ n -1) has at least one 1-digit under one 0-digit of n and so n k is dividable by p. Hence the right member of Equation ( 1) is 0 since the field has characteristic p.

• If n is not a power of p, it is written 1d 1 . . . d β 1e 1 . . . e γ in base p and choosing for k the number 10 γ in base p, n k is not dividable by p. Therefore the coefficient of X k in Equation ( 1) is not 0, so the right-hand side of (1) is not the null polynomial.

Hence L is the non-recognizable language of powers of p and the support of S is not recognizable.

Note that this example shows that F p (X) is a non-SR-semiring. It is an example of a non-SR-semiring which does not include Z, answering the question of the existence of such semirings [START_REF] Kirsten | Recognizability of the support of recognizable series over the semiring of the integers is undecidable[END_REF].

Figure 1 :

 1 Figure 1: An automaton over Fp(X) and {a} whose behaviour has non-recognizable support.

Acknowledgements. We thank an anonymous referee whose fruitful comments improved the presentation of our results.