An Algebraic Interpretation of the Multiplicity Sequence of an Algebraic Branch


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1 An Algebraic Interpretation of the Multiplicity Sequence of an Algebraic Branch Une Interprétation Algébrique de la Suite des Ordres de Multiplicité d une Branche Algébrique Proc. London Math. Soc. (2), 50 (1948), by Cahit Arf [Received 7 September Read 15 November Received in revised form 30 September 1946] Translated by: Ali Sinan Sertöz in The sequence of multiplicities of successive points of an algebraic branch can (p 256) 1 be defined through purely algebraic notions. In what follows we present such a definition which does not differ from the geometric definition except by its form. We hope that this definition will constitute an answer to a question posed by P. Du Val on the relation which exists between his results and the power series 2 expansion of the branch under consideration. Section 1: 3 k being any field, we consider a ring H formed by some power series of a single variable t with coefficients in k. Let W (H) = {i 0 = 0, i 1, i 2,..., i r, i r+1,... } 4 P. Du Val, The Jacobian algorithm and the multiplicity sequence of an algebraic branch, Rev. Faculté Sci. Univ. Istanbul (Série A), 7 (1942),
2 be the orders (i.e. the degrees of the first terms with nonzero coefficients) of the elements of H. The integers i 0, i 1,...,, i r,... form a semigroup of the nonnegative 5 integers. S 0, S i1, S i2,..., S ir,... being elements of H of orders i 0, i 1,..., i r,... respectively, any element of this ring is of the form α l S il (α l k). l=0 We assume that H contains all the series of this form. We denote by I h the set of elements of H of orders larger than or equal to h. I h is clearly an ideal of H and its elements are of the form α l S il (α l k). i l h Lemma 1. 6 ν being the gcd of the elements of W (H), for r sufficiently large, (p 257) one has i r+1 = i r + ν, i r+2 = i r + 2ν,..., i r+l = i r + lν,... and there exists a power series of order 1, ( ) τ = t 1 + δ l t l l=1 (δ l k) such that every element of H is of the form j=0 α jτ jν. Proof. Let us denote the gcd of the integers i 1, i 2,..., i l by ν l. Each of these numbers divides all those that come before it. It follows that for ρ sufficiently large we have ν ρ = ν ρ+1 = ν ρ+2 = = ν. Then let ν = m 1 i 1 + m 2 i m ρ i ρ, m 1, m 2,..., m ρ being integers which are positive, zero or negative. m being the largest of the integers m h (i 1 /ν 1), the multiples of ν which are greater than i = mi 1 + mi mi ρ are contained in W (H). In fact we have, for l = 0, 1, 2..., i 1 /ν 1, i + lν = (m + lm 1 )i 1 + (m + lm 2 )i (m + lm ρ )i ρ = n 1 i 1 + n 2 i n ρ i ρ, 2
3 with n h 0; since m m h l. For l = i 1 /ν, we have i + i 1 W (H). In general, the multiples of ν which are greater than i can be written in the form i + ji 1 + lν (l = 0, 1, 2,..., i 1 /ν 1, j 0) and it is obvious that all of ρ these integers are of the form n h i h with n h 0; i.e. belong to W (H). S i1 = l=i 1 σ l t l (σ l k, σ i1 h=1 choose a power series of the form τ = t 0) 7 being an element of order i 1 in H, we can ( 1 + ) δ l t l, (δ l k) in such a way that we have S i1 = σ i1 τ i 1. Under these conditions the power series in t with coefficients in k may be written in the form of power series in τ with coefficients in k. In particular the elements of H can be written in the form α jν τ jν. It suffices to prove this for the elements of H of orders greater than j=0 j=nν i; since every element of H can be considered as a quotient of an element in H of order greater than i by a suitably chosen power of S i1 = σ i1 τ ν(i1/ν). The orders of the elements of H being multiples of ν, any element of H is of the form α j τ j (α j k, α Nν 0). For Nν i, the ring H contains the elements, S Nν+ν, S Nν+2ν,..., ( S Nν+lν = j=nν+lν of orders Nν + ν, Nν + 2ν,... respectively. β l S Nν+lν in such a way that the difference l=1 l=1 α l,j τ j, α l,j k, α l,nν+lν 0 ) We can then choose the series (p 258) S Nν = α j τ j j=nν β l S Nν+lν = α Nν τ Nν + α µ τ µ + l=1 does not contain any terms of order divisible by ν, other than the first. Indeed suppose that β 1, β 2,..., β h are chosen such that the terms of orders Nν +ν, Nν + 2ν,..., Nν + hν of the difference j=nν α j τ j h l=1 β l S Nν+lν = α Nν τ Nν + α (h) µ h τ µ h + 3
4 vanish; it suffices then to set β h+1 = α(h) Nν+hν+ν α h+1,nν+hν+ν so that the terms of orders Nν + ν, Nν + 2ν,..., Nν + hν, Nν + hν + ν of the difference j=nν h+1 α j τ j β l S Nν+lν = α Nν τ Nν + α µ (h+1) h+1 τ µ h+1 + l=1 vanish. Under these conditions the series S Nν reduces to α Nν τ Nν. Otherwise the difference ( ) N S i 1/ν Nν αi 1/ν Si1 Nν = i 1 ν αi 1/ν 1 Nν α µ τ Nν(i1/ν 1)+µ + σ i1 whose order is not divisible by ν will be in H. Therefore every element of H of order greater then i is a linear combination with coefficients in k of elements of the form α Nν τ Nν = S Nν. 8 Remark. After the preceding theorem, the ring H may be considered as a subring of the ring of power series of the variable T = τ ν with coefficients in k. Let us set i h = i h /ν. The orders of the elements of H with respect to this new variable will be i 0 = 0, i 1, i 2,..., i r,..., and for r sufficiently large, one will have i r+1 = i r + 1, i r+2 = i r + 2,.... Lemma 2. The inverse of every element of order zero of H is also an element of H. Proof. If the order of a = α h S ih is zero, then α 0 is different than zero. In fact h=0 the coefficients β h of the product α 1 0 (1 + β h S ih ) h=1 can be chosen such that we have (p 259) 4
5 Section 2: aα0 1 h=1 n (1 + β h S ih ) 1 mod t in+1. Suppose now that this choice has been made for β 1, β 2,..., β n 1. We have n 1 aα0 1 h=1 and it suffices to set β n = γ n to have (1 + β h S ih ) = 1 + γ n S in + γ n+1 S in+1 + aα 1 0 n (1 + β h S ih ) 1 mod t in+1. h=1 For the coefficients β h chosen in this manner we obviously have Remark. h=0 aα 1 0 (1 + β h S ih ) = 1. h=1 α h S ih being an element of order zero in H, to each nth root of α 0 contained in k corresponds an nth root of proof of this fact is similar to that of Lemma 2. α h S ih contained in H. The Lemma 3. If one denotes by I h /S h the set of quotients of elements of I h by S h, and by [I h /S h ] the ring generated by I h /S h, the ring [I h /S h ] does not depend on the choice of S h among the elements of H of order h. h=0 Proof. Let us first note that the set I h /S h contains the ring H and consequently [I h /S h ] H. Let S h = ɛs h be another element of order h in H. ɛ is then an element of [I h /S h ]. It follows from Lemma 2 that ɛ 1 is also an element of [I h /S h ]. We then have I h /S h = I h /ɛs h = ɛ 1 (I h /S h ) [I h /S h ] 5
6 and therefore [I h /S h] [I h /S h ]. We can obviously show in exactly the same manner that we also have [I h /S h ] [I h /S h]. We then have [I h /S h ] = [I h/s h ]. The ring [I h /S h ] being independent of the choice of S h, we can denote it by [I h ]. Remark. The semigroup W ([I h ]) clearly contains the semigroup generated by the integers i h i h = 0, i h+1 i h, i h+2 i h,... which are the orders of the elements of I h /S ih. But as the following example (p 260) shows, W ([I ih ]) is not necessarily equal to this semigroup: Let us consider the ring H formed by all series of the form α ij X i Y j (α ij k), i,j 0 where X = t 4, Y = t 10 + t 15. One easily shows that W (H) is formed by the integers 0,4, 8, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26 28, 29, 30, 32, 33, 34, 35, 36, 37, 38,.... Then the orders of the elements of I 4 /X are the integers which generate the semigroup 0, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34,... 0,4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,... while [I 4 ] contains the element (Y/X) 2 X 2 = 2t 17 + t 22 whose order is 17. 6
7 Remark. If for some particular choice of S h, the ring [I h ] is equal to I h /S h, then it is the same for all choices of S h. In fact, S h = ɛs h being another element of order h of H, we have I h /S h = ɛ 1 I h /S h = ɛ 1 [I h ] = [I h ]; since every element S of [I h ] is equal to an element ɛs of [I h ] multiplied by ɛ 1. Definition. We say that the ring H is canonical 9 if one has [I h ] = I h /S h for all h W (H). Remark. If H is a canonical ring, the integers i h i h = 0, i h+1 i h, i h+2 i h,... form a semigroup for every h. A semigroup of nonnegative integers is called canonical if the sequence i 0 = 0, i 1, i 2,..., i h,... i h i h = 0, i h+1 i h, i h+2 i h,... is a semigroup for each h. If the sequence of increasing integers i 0 = 0, i 1, i 2,..., i h,... is a canonical semigroup, then the power series α h t i h h=0 (α h k), clearly forms a canonical ring. W (H) can be canonical without it being the case (p 261) for H: The ring formed by the series of the form α ijl X i Y j Z l (α ijl k), i,j,l 0 with X = t 4, Y = t 10 + t 15, Z = t 27, is such that the orders 0, 4, 8, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30,... of its elements form, as one can easily verify, a canonical semigroup, while H is not a canonical ring, since [I 4 ] an element of order 17, (Y/X) 2 X 3 = 2t 17 + t 22, which is not contained in I 4 /X. 7
8 Lemma 4. The intersection of several canonical rings is canonical. Proof. It obviously suffices to prove the lemma only for the intersection of two canonical rings. H and H being two canonical rings, let S be a common element of these two rings. Let h be the order of S let I h and I h be the set of elements of H and H whose orders are not less than h. It suffices to show that (I h I h)/s = I h /S I h/s is a ring. Now I h /S and I h /S being rings, it is the same for their intersection. Remark. If H is a canonical ring, then so is [I ih ]. Indeed consider the set of elements of I ih. These elements are of the form α ν S iν ν=h (α ν k). H being a canonical ring, the ring [I ih ] consists of the set of series of the form S iν α ν whose orders are the numbers S ih ν=h 0, j 1 = i h+1 i h, j 2 = i h+2 i h,.... The set of elements of [I ih ] of order greater than or equal to j l is then the set of series of the form S iν α ν (α ν k). S ih ν=h+l ν=h+l S ih+l /S ih being an element of order j l = i h+l i h of this set, the set of elements ( )/ S iν Sih+l S iν α ν = α ν S ih S ih S ih+l is the ring [I ih+l ]. ν=h+l if N being the set of all nonnegative integers, we show in a similar manner that (p 262) {0, i 1, i 2,..., i r + Nν} In what follows N will always denote the set of all nonnegative integers. 10 8
9 is a canonical semigroup, it is the same for {0, i h+1 i h,..., i r i h + Nν}. Remark. If the integers i 0 = 0, i 1, i 2,..., i h,... form a canonical semigroup, then we have i h+1 i h i h i h 1. In fact, before the integers i h 1 i h 1 = 0, i h i h 1, i h+1 i h 1,..., i r i h 1,... form a semigroup, we must have i h+1 i h 1 2(i h i h 1 ); from which the inequality i h+1 i h i h i h 1 follows immediately. Section 3: From the remark which follows immediately Lemma 1, I (N 1)ν contains all the power series whose orders in T = τ ν are greater than or equal to N 1 provided that N is sufficiently large. [I (N 1)ν ] is then the ring k[t ] of all the power series in T with coefficients in k. This remark leads to the following construction which allows us to obtain all the canonical rings as well as all the canonical semigroups. We begin by considering the ring [I (N 1)ν ] = k[t ] of all power series in T and the semigroup Nν of multiples of ν by nonnegative integers. We choose an element T r 1 of nonzero order in [I (N 1)ν ], and a nonzero element ν r 1 (= w(t r 1 ) ) in Nν and we set [I ir 1 ] = k + T r 1 [I (N 1)ν ] (i r = (N 1)ν). The ring [I ir 1 ] and the semigroup {0, ν r 1 + Nν} (= W ([I ir 1 ])) are canonical. Similarly we choose an element T r 2 of nonzero order in [I ir 1 ] and a positive integer ν r 2 (= w(t r 1 )) in {0, ν r 1 + Nν}, and we set [I ir 2 ] = k + T r 2 [I ir 1 ] = k + kt r 2 + T r 2 T r 1 k[t ], W ([I ir 2 ]) = {0, ν r 2, ν r 2 + ν r 1 + Nν}. In what follows w α i t i denotes the order of the series α i t i in t. i=µ 9 i=µ
10 Thus we obtain a new canonical ring and also a canonical semigroup. Continuing in this manner we finally obtain the canonical ring k + kt 1 + kt 1 T kt 1 T 2 T r 2 + k[t ]T 1 T 2 T r 2 T r 1 and the canonical semigroup (p 263) with {0, ν 1, ν 1 + ν 2,..., ν 1 + ν ν r 1 + Nν} T h kt h+1 + kt h+1 T h k[t ]T h+1 T h+2 T r 1, (w(t h ) =) ν h {ν h+1, ν h+1 + ν h+2,..., ν h+1 + ν h ν r 1 + Nν}. Section 4: Given a ring H, the intersection of all canonical rings containing H is a canonical ring H which we call the canonical closure 11 of H. Similarly G = {0, i 1, i 2,..., i r 1 + Nν} being a semigroup of nonnegative integers (ν = (i 1, i 2,..., i r 1 + ν)), the intersection of all canonical semigroups containing G is a canonical semigroup G; we call it the canonical closure of G. It follows from this definition that W ( H) contains the canonical semigroup W (H); but these two semigroups 12 are not necessarily equal, since W (H) may be canonical without H being so. Section 5: Given a semigroup G = {0, i 1, i 2,..., i r 1 + Nν} (ν = (i 1, i 2,..., i r 1, i r 1 + ν)), the canonical closure G of G is obtained as follows: We consider the semigroup {0, i 1 + G 1 } where G 1 is the semigroup of integers of the form α 2 (i 2 i 1 ) + α 3 (i 3 i 1 ) + + α n (i n i 1 ), where the coefficients α 1, α 2,..., α n are nonnegative integers. The semigroup {0, i 1 + G 1 } which then contains G is obviously contained in G. Note that the elements of G 1 which are less than i h+1 i 1 are of the form α 2 (i 2 i 1 ) + α 3 (i 3 i 1 ) + + α h (i h i 1 ); 10
11 the integers α 2 (i 2 i 1 ) + α 3 (i 3 i 1 ) + + α n (i n i 1 ) with n h + 1, α n 0 are in fact greater than or equal to i h+1 i 1. In particular the smallest element of G 1 is i 2 i 1. Furthermore it follows that the elements of {0, i 1 + G 1 } which are less than i h+1 depend only on i 1, i 2,..., i h, and are linear combinations of these with integer coefficients. The semigroup {i 1 + G 1 } being contained in G, it is the same for {i 1 + G 1 } which contains {i 1 + G 1 } G, and is canonical. The construction of G is thus reduced to the construction of the canonical closure of a semigroup of the form G 1 = {0, i 1, i 2,..., i r 1 + Nν}; for which we have i r 1 i r 1 i 1. The repetition of this construction reduces the proposed construction to that of the canonical closure of a semigroup G N which itself reduces, for N sufficiently large, to the semigroup Nν of all nonnegative (p 264) multiples of ν. Nν being its own canonical closure, the proposed procedure thus terminates. Note that the elements of G which are thus constructed depend only on the elements of G which are not greater than themselves; and they are linear combinations of them with integer coefficients. Suppose in fact that this is proved for the closure G 1 of G 1. The elements of G 1 which are smaller than i h+1 i h depend only on the elements of G 1 which are smaller than i h+1 i h, and they are their linear combinations with integer coefficients; now these latter ones in turn depend only on i 1, i 2,..., i h and are their linear combinations with integer coefficients. It follows that the elements of {0, i 1 + G 1 } = G which are smaller than i h+1 i h depend only on i 1, i 2,..., i h, and they are their linear combinations with integer coefficients. Given a canonical semigroup G = {0, i 1, i 2,..., i r 1 + Nν} (ν = (i 1, i 2,..., i r 1, i r 1 + ν)), there exist only a finite number semigroups g such that g = G. In fact let g = {0, j 1, j 2,..., j s, j s+1,... } be such a semigroup. Let j 1, j 2,..., j n of the integers j 1, j 2,..., j s,... be smaller than i r+1 = i r 1 + 2ν. Since i r 1 and i r 1 + ν are linear combinations of j 1, j 2,..., j n with integer coefficients, the gcd of these numbers is ν. Now to each system of positive integers smaller than i r+1 = i r 1 + 2ν whose gcd is ν, we 11
12 can associate a multiple jν of ν such that every semigroup of nonnegative integers containing the system, contains all the multiples of ν larger than jν. Let Lν be the largest of the integers jν which are thus associated to systems of positive multiples of ν smaller than i r+1 = i r 1 + 2ν. The semigroups g for which we have g = G contain then all the multiples of ν which are larger than Lν and they differ among themselves only by those elements which are smaller than Lν. Theorem 1. The intersection of all the semigroups g such that g = G is a semigroup g χ such that g χ = G. Proof. Let g be a semigroup such that we have g = G and that no proper sub semigroup of g has this property; we will show that g = g χ. Let i be the smallest element of g not in g χ. Let i 0 = 0, i 1, i 2,..., i h be the elements of g and g χ which are smaller than i. Since i is not contained in g χ, the number i is not of the form α 1 i 1 + α 2 i α h i h, where α 1, α 2,..., α h are nonnegative integers. On the other hand g χ being the intersection of all semigroups whose canonical closure is G, there exists a semigroup g such that g = G and which does not contain the number i. Since the elements of G = g which are smaller than i depend only on i 1, i 2,..., i h, the (p 265) semigroup g obtained by removing from g all the the positive integers smaller than i except i 1, i 2,..., i h still has the property that g = G. It follows that the elements of G which are smaller than or equal to i depend only on the numbers i 1, i 2,..., i h ; since g does not contain the number i. Therefore the canonical closure of the subsemigroup of g obtained by removing from it the number i is still equal to G. This contradicts the choice of g. We then have g χ = g and and consequently g χ = G. The semigroup g χ defined in the statement of Theorem 1 is called the characteristic subsemigroup of all the g such that g = G. It is clear that the semigroup g χ is such that every proper subsemigroup of g χ has a canonical closure different than g χ = G. Conversely if g χ is such that for every subsemigroup g of g χ we have g g χ, then g χ is its own characteristic subsemigroup. g χ = {0, i 1, i 2,..., i r 1, i r,... } being the characteristic subsemigroup of g, let us consider the integers χ 1, χ 2,..., χ h defined in the following manner: χ 1 = i 1 ; χ 2 is the smallest of the integers i 1, i 2,..., i r,... which is not of the form α 1 χ 1 where α 1 is a nonnegative integer; χ 3 is the smallest of the integers 12
13 i 1, i 2,..., i r,... which is not of the form α 1 χ 1 + α 2 χ 2 where α 1, α 2 are nonnegative integers; finally χ 1, χ 2,..., χ n being defined, χ n+1 is the smallest of the integers i 1, i 2,..., i r,... which is not of the form α 1 χ 1 + α 2 χ α n χ n where α 1, α 2,..., α n are nonnegative integers. The numbers χ 1, χ 2,..., χ h defined in this manner are called the characters of g. Theorem 2. γ 1 < γ 2 < < γ l being a collection of positive integers, the set of characters of the semigroup g of integers of the form α 1 γ 1 + α 2 γ α l γ l, where α 1, α 2,..., α l are nonnegative integers, is contained in the collection γ 1, γ 2,..., γ l. Proof. Let χ i be the smallest of the characters χ 1, χ 2,..., χ h of g which is not contained in the set γ 1, γ 2,..., γ l. It follows from the definition of g χ that χ i is of the form α 1 γ 1 + α 2 γ α l γ l where α 1, α 2,..., α l are nonnegative integers and where γ 1, γ 2,..., γ l are those integers among γ 1, γ 2,..., γ l which are smaller than χ i. Since γ 1, γ 2,..., γ l are elements of the canonical closure of g χ, every semigroup containing χ 1, χ 2,..., χ i 1 contains also γ 1, γ 2,..., γ l. This implies that the canonical closure of the semigroup of linear combinations with nonnegative integer coefficients of χ 1, χ 2,..., χ i 1, χ i+1,..., χ h contains χ i, and it follows that g χ is not a characteristic semigroup. Therefore the set γ 1, γ 2,..., γ l necessarily contains the set χ 1, χ 2,..., χ h. Theorem 3. g being the semigroup of linear combinations of [the integers] 13 0 < (p 266) γ 1 < γ 2 < < γ l with nonnegative integer coefficients, the integers with the property that ν 1, ν 2,..., ν N 2, ν N 1, ν g = {0, ν 1, ν 1 + ν 2,..., ν 1 + ν ν N 1 + Nν}, are obtained from γ 1, γ 2,..., γ l by the quasijacobian algorithm of Du Val. The integers ν 1, ν 2,..., ν N 1, ν appear there as divisors, while the partial quotients Du Val, loc. cit. 13
14 represent the number of times each divisor appears in the sequence ν 1, ν 2,..., ν N 1, ν. Conversely if the numbers ν 1, ν 2,..., ν N 1, ν are obtained from γ 1, γ 2,..., γ l by the quasijacobian algorithm of Du Val, the partial quotients being the number of times each divisor appear in the sequence ν 1, ν 2,..., ν N 1, ν, then the canonical closure of the semigroup of the integers of the form α 1 γ 1 + α 2 γ α l γ l, where α 1, α 2,..., α l are nonnegative integers, is g. Proof. ν being the greatest common divisor of the elements of g, we have γ 1 ν. If γ 1 = ν, the semigroup g consists of the set of all multiples of γ 1 = ν, and we have g = g = {Nν}. In this case the algorithm terminates at the first step. Let s assume that the proposition is proved for every set γ 1, γ 2,,..., γ l for which γ 1 < γ 1, and prove it for for the set γ 1, γ 2,..., γ l. Let γ i be the smallest of the integers γ 1, γ 2,..., γ l which is not divisible by γ 1. Let q be the quotient of γ i by γ 1 and let us consider the semigroup Γ of linear combinations of γ i qγ 1, γ i+1 qγ 1,..., γ l qγ 1, γ 1 with nonnegative integer coefficients. The semigroup g clearly contains the semigroup {0, γ 1, 2γ 1,..., qγ 1 + Γ} which contains g. We then have g = {0, γ 1, 2γ 1,..., qγ 1 + Γ}, ν 1 = γ 1, ν 2 = γ 1,..., ν q = γ 1 i.e. Γ = {0, ν q+1, ν q+1 + ν q+2,..., ν q ν N 1 + Nν}. The integers γ i qγ 1, γ i+1 qγ 1,..., γ l qγ 1, γ 1 being the remainders of the (i 1)st division of the algorithm applied to the numbers γ 1, γ 2,..., γ l, it suffices to show that the integers ν q+1, ν q+2,..., ν N 1, ν are obtained by applying the algorithm to the integers γ i qγ 1, γ i+1 qγ 1,..., γ l qγ 1, γ 1. Now γ i qγ 1 being smaller than γ 1, this was assumed done. Conversely, if the numbers (p 267) ν 1, ν 2,..., ν N 1, ν are obtained from γ 1, γ 2,..., γ l by the quasijacobian algorithm of Du Val, the canonical closure of the semigroup of linear combinations of γ 1, γ 2,..., γ l where coefficients are nonnegative integers is {0, ν 1, ν 1 + ν 2,..., ν 1 + ν ν N 1 + Nν}, 14
15 which follows from the proposition we have just proved. χ 1, χ 2,..., χ h being the characters of g, the semigroup of linear combinations of χ 1, χ 2,..., χ h with coefficients being nonnegative integers is the characteristic sub semigroup g χ of g. It follows from theorems 3 and 2 that the integers ν 1, ν 2,..., ν N 1 are obtained from the characters of g by the quasijacobian algorithm of Du Val, and all the systems of nonnegative integers γ 1, γ 2,..., γ l for which the algorithm produces the same result are obtained from the system of characters of g by adjoining to it numbers arbitrarily chosen from g. Section 6: Now let us consider a ring H and its canonical closure H. The ring H being of the form H = k + ks i1 + ks i2 + + k[t ]S ih its canonical closure H can be constructed as follows: Denote by H 1 the ring [I i1 ] = ( ) α2 ( ) α3 ( ) αh 1 Si2 Si3 Sih 1 k + k[t ] S i h, S i1 S i1 S i1 where the summation is over all exponent systems of nonnegative integers α 2, α 3,..., α h 1 such that α 2 (i 2 i 1 ) + α 3 (i 3 i 1 ) + + α h 1 (i h 1 i 1 ) is less than i h i 1. The canonical closure H of H clearly contains S i1 k + H 1 S i1 which contains H and we have H = k + H 1 S i1, where we denoted by H 1 the canonical closure of H 1. In general, H i being defined, denote by H i+1 the ring obtained from H i in the same way H 1 is obtained from H. It is clear that for N sufficiently large, H N is isomorphic to K[T ]. Let T i+1 be an element of positive order in H i. Then we obviously have H = k + kt 1 + H 2 T 1 T 2 (with T = S i ) = k + kt 1 + kt 1 T 2 + H 3 T 1 T 2 T 3 = k + kt 1 + kt 1 T kt 1 T 2 T N 1 + H N T 1 T 2 T N 1 T N = k + kt 1 + kt 1 T kt 1 T 2 T N 1 + k[t ]T 1 T 2 T N 1 T N. 15
16 Remark. For any integer n, the ring k + H 1 S i1 mod t n depends only on H mod t n. To prove this it suffices to show that H 1 mod t n i 1 depends only on H (p 268) mod t n.similarly the ring k + H 2 T 2 mod t n i 2 depends only on H 1 mod t n i 1. The ring k+kt 1 +H 2 T 1 T 2 mod t n then depends only on H mod t n. Continuing in this manner we obtain eventually that H = k + kt 1 + kt 1 T kt 1 T 2 T N 1 + H N T 1 T 2 T N mod t n depends only on H mod t n. Lemma 5. If H mod t n is equal to H mod t n, then the set H mod t n+1 is equal to one of the sets k + ks i1 + ks i2 + + ks il 1 + [I il ]S il mod t n+1 (i l < n + 1). Proof. The set H mod t n being the same as H mod t n, the set H mod t n+1, which contains the set H mod t n+1, consists of the elements of the form S + α S n mod t n+1 where S is an element of H, S n a fixed element of order n belonging to H, and α an element of k. Hence every ring H mod t n+1, contained in the ring H mod t n+1 is identical to H mod t n+1, if it is contained in H mod t n+1 without it being identical. Consider now the ring k + S i1 [I i1 ] mod t n+1 which contains H mod t n+1 and which is contained in H mod t n+1. After what we have just noted, the ring k + S i1 [I i1 ] mod t n+1 is identical to one of the two rings H mod t n+1, H mod t n+1. If it is not identical to the first, we have [I i1 ] = I i1 /S i1 mod t n+1 i 1. As [I i1 ] mod t n+1 i 1 depends only on [I i1 ] mod t n+1 i 1, the sets [I i1 ] mod t n+1 i 1, k + S i 2 S i1 [I i2 ] mod t n+1 i 1 will be identical, since I i2 /S i1 is the set of elements of positive order in I i1 /S i1. It follows that H mod t n+1 is identical to one of the rings k + S i1 [I i1 ] mod t n+1, k + ks i1 + [I i2 ]S i2 mod t n+1. 16
17 If H mod t n+1 is neither identical to k + S i1 [I i1 ] mod t n+1 nor to k + ks i1 + [I i2 ]S i2 mod t n+1 these two rings are identical to H mod t n+1. Under these conditions we have [I i2 ] I i2 /S i2 mod t n+1 i 2, from which we can conclude the identity of the two sets [I i2 ] mod t n+1 i 2, k + [I i2 ] S i 2 S i1 mod t n+1 i 2. H mod t n+1 is then identical to one of the sets (p 269) k + [I i1 ]S i1 mod t n+1, k + ks i1 + [I i2 ]S i2 mod t n+1, k + ks i1 + ks i2 + [I i3 ]S i3 mod t n+1. Continuing in this manner we can show that H mod t n+1 is identical to one of the sets k + [I i1 ]S i1 mod t n+1 k + ks i1 + [I i2 ]S i2 mod t n k + ks i1 + ks i2 + + [I il ]S il mod t n+1 k + ks i1 + ks i2 + + ks il + [I il+1 ]S il+1 mod t n+1. Now for i l+1 n + 1, the last one of these sets is H mod t n+1. Then H mod t n+1 is identical to one of the sets k + ks i1 + ks i2 + + [I il ]S il mod t n+1 for i l n. X 1, X 2,..., X n being power series in t with positive orders, we denote by k[x 1, X 2,..., X n ] the ring formed by the series of the form αj1 j 2 j n X j 1 1 X j 2 2 X jn n where α j1 j 2 j n k and the summation is over all systems of nonnegative integers j 1, j 2,... j n. 17
18 Lemma 6. The elements Y 1, Y 2,..., Y ν of H being chosen such that w(y j ) is the smallest element of W ( H) which is not contained in W (k[y 1, Y 2,..., Y j 1 ]), if the elements Y 1, Y 2,..., Y ν 1 are respectively congruent to Y 1, Y 2,..., Y ν 1 mod t w(yν), then the smallest element of W ( H) not contained in W (k[y 1, Y 2,..., Y ν 1]) is w(y ν ). Proof. The rings H mod t w(yν), k[y 1, Y 2,..., Y ν 1 ] mod t w(yν), k[y 1,..., Y ν 1] mod t w(yν) being clearly identical, it suffices to show that k[y 1, Y 2,..., Y ν 1] does not contain an element of order w(y ν ). Every element of k[y 1, Y 2,..., Y ν 1] mod t w(yν)+1 being of the form P (Y 1, Y 2,..., Y ν 1) mod t w(yν)+1 where P (Y 1, Y 2,..., Y ν 1) is a polynomial with coefficients in k, it suffices to show that w(p (Y 1, Y 2,..., Y ν 1)) cannot be equal to w(y ν ). If the polynomial P (Y 1, Y 2,..., Y ν 1) contains a [nonzero] 14 constant term, then Y 1, Y 2,..., Y ν 1 being elements of positive order we have w(p (Y 1, Y 2,..., Y ν 1)) = 0 w(y ν ). If P (Y 1, Y 2,..., Y ν 1) contains terms of degree 1 without containing a [nonzero] constant term, then we (p 270) can write it in the form P 1 (Y 1, Y 2,..., Y j 1) + βy j + P 2 (Y 1, Y 2,..., Y ν 1) with β 0; P 2 (Y 1, Y 2,..., Y ν 1) being the sum of terms of positive degree with respect to the set Y j, Y j+1,..., Y ν 1 except the term βy j. w(p 2 (Y 1, Y 2,..., Y ν 1)) is then greater than w(y j ) which is by definition different than the order of We then have P 1 (Y 1, Y 2,..., Y j 1) P 1 (Y 1, Y 2,..., Y j 1 ) mod t w(yν). w(p (Y 1, Y 2,..., Y ν 1)) = min(w(y j )), w(p 1 (Y 1, Y 2,..., Y j 1)) < w(y ν ). Finally if P (Y 1, Y 2,..., Y ν 1) contains neither a term of degree 1 nor of degree 0, then we can write P (Y 1, Y 2,..., Y ν 1) P (Y 1, Y 2,..., Y ν 1 ) mod t w(yν)+1. 18
19 w(p (Y 1, Y 2,..., Y ν 1 )) being different than w(y ν ), it is the same for w(p (Y 1, Y 2,..., Y ν 1)). Lemma 7. Y 1, Y 2,..., Y ν 1, Y ν and Y 1, Y 2,..., Y ν 1 having the same properties as in the statement of Lemma 6, if the canonical closure of k[y 1, Y 2,..., Y ν 1 ] does not contain an element of order w(y ν ), then it is the same for the canonical closure of k[y 1, Y 2,..., Y ν 1]. Proof. Let i 0 = 0, i 1, i 2,..., i µ,... be the orders of the elements of k[y 1, Y 2,..., Y ν 1] written in increasing order and let I i µ be the set of elements of k[y 1, Y 2,..., Y ν 1] whose orders are not smaller than i µ. Denote by S i l an element of order i l of k[y 1, Y 2,..., Y ν 1], and by H 15 the canonical closure of k[y 1, Y 2,..., Y ν 1]. The rings H mod t w(yν), H mod t w(yν), k[y 1, Y 2,..., Y ν 1] mod t w(yν) mod t w(yν)+1 is iden being identical, it follows from Lemma 5 that the ring H tical to one of the rings k + ks i 1 + ks i [I i l ]S i l mod t w(yν)+1 with i l < w(y ν ). Let µ be the smallest of these integers l for which this identity holds. If µ = 0, then H mod t w(yν)+1 is identical to k[y 1, Y 2,..., Y ν 1] mod t w(yν)+1 which does not contain an element of order w(y ν ). Suppose then that µ is positive. To show that H does not contain an element of order w(y ν ), it suffices to show that [I i µ ] does not contain an element of order w(y ν ) i µ. Let I iµ and I iµ be the sets of elements of order not smaller than i µ of k[y 1,..., Y ν 1 ] and H. The rings H mod t w(yν), k[y 1, Y 2,..., Y ν 1 ] mod t w(yν), k[y 1, Y 2,..., Y ν 1] mod t w(yν) being identical, it is the same for the sets (p 271) [ I iµ ] mod t w(yν) iµ I iµ /S iµ mod t w(yν) iµ, I i µ /S i µ mod t w(yν) iµ, 19
20 where S iµ is an element of k[y 1, Y 2,..., Y ν 1 ], such that we have S iµ S i µ mod t w(yν). It follows that we can associate to every element Z of I i µ /S i µ I iµ /S iµ in such a way that we have an element Z of Z = Z mod t w(yν) iµ. Let us consider in particular a set of elements Z 1, Z 2,..., Z ρ of I i µ /S i µ chosen in the following way: (1) Z 1 is an element of smallest positive order in I i µ /S i µ, (2) Z 1, Z 2,..., Z j 1 being chosen, we choose Z j in such a way that w(z j) is the smallest positive element of W (I i µ /S i µ ) which is not contained in W (k[z 1, Z 2,..., Z j 1]), (3) w(z ρ) < w(y ν ) i µ + 1 and every element of W (I i µ /S i µ ) smaller than w(y ν ) i µ + 1 is contained in W (k[z 1, Z 2,..., Z ρ]). k[y 1, Y 2,..., Y ν 1] mod t w(yν)+1 being distinct than H mod t w(yν)+1 while k[y 1, Y 2,..., Y ν 1] mod t w(yν)+1 is identical to H mod t w(yν), the ring k[y 1, Y 2,..., Y ν 1] cannot contain elements of orders w(y ν ). It follows that the numbers w(z 1), w(z 2),..., w(z ρ) are smaller than w(y ν ) i µ.the conditions imposed on the choice of Z 1, Z 2,..., Z ρ implies further the identity of the rings [I i µ ] mod t w(yν) iµ+1, k[z 1, Z 2,..., Z ρ] mod t w(yν) iµ+1 ; It suffices then to show that k[z 1, Z 2,..., Z ρ] does not contain an element of order w(y ν ) i µ. Now let Z 1, Z 2,..., Z ρ be elements of I iµ /S iµ such that we have Z j Z j mod t w(yν) iµ (j = 1, 2,..., ρ). The canonical closure of k[y 1, Y 2,..., Y ν 1 ] not containing any element of order w(y ν ), the ring k[z 1, Z 2,..., Z ρ ] does not contain any element of order w(y ν ) i µ. The elements Z 1, Z 2,..., Z ρ, Z ρ+1 = Y ν /S iµ of [ I iµ ] and Z 1, Z 2,..., Z ρ fulfill the conditions of the statement of Lemma 6 with respect to the canonical ring [ I iµ ]. The ring k[z 1, Z 2,..., Z ρ] then cannot contain elements of order w(z ρ+1 ) = w(y ν ) i µ. Let us now consider a set of elements X 1, X 2,..., X m of H chosen as follows: X 1 is an element of smallest positive order in H; X 1, X 2,..., X l 1 being chosen, X l is an element of H such that w(x l ) is the smallest element of 20
21 W ( H) which is not contained in W (H l 1 ), where H l 1 16 is the canonical clo (p 272) sure of k[x 1, X 2,..., X l 1 ]. The elements of W ( H) being linear combinations with nonnegative integer coefficients of some finite number of elements, the elements X 1, X 2,..., X l,... chosen in this manner can only be finite. A set of such elements (X 1, X 2,..., X m ) will be called in what follows a base of H. Theorem 4. (X 1, X 2,..., X m ) being a base of H, the integers w(x 1 ), w(x 2 ),..., w(x m ) depend on H and they constitute a subset of the characters of H. Let us first prove the following proposition which will facilitate the proof of this theorem. Lemma 8. H l being the canonical closure of k[x 1, X 2,..., X l ] where X 1, X 2,..., X m is a base of H, one can choose the elements Y 1, Y 2,..., Y ν,... of H l satisfying the conditions of the statement of Lemma 6 considered for the ring H l (i.e. w(y j ) is the smallest element of w(h l ) not contained in W (k[y 1, Y 2,..., Y j 1 ])) in such a manner that the sequence Y 1, Y 2,..., Y ν,... contains the set X 1, X 2,..., X l. Proof. For l = 1, we clearly have H 1 = k[x 1 ] and we can set Y 1 = X 1. Assume that the proposition is proved for l and let us prove it for l + 1. Let Y 1, Y 2,..., Y ν be the elements chosen from H l whose orders are smaller than w(x l+1 ). The elements of W (H l ) which are smaller than w(x l+1 ) are then the same as those of W (k[y 1, Y 2,..., Y ν ]). The smallest element of W (H l+1 ) not contained in W (H l ) being w(x l+1 ), set Y ν+1 = X l+1, and choose Y ν+2, Y ν+3,... from H l+1 in accordance with the statement of Lemma 6 with respect to H l+1. The sequence Y 1, Y 2,..., Y ν, Y ν+1,... then satisfies for H l+1 the conditions of the statement of the proposition which we wanted to prove. Proof of Theorem 4. Let X 1, X 2,..., X m and X 1, X 2,..., X m be two bases of H. If the integers w(x 1 ), w(x 2 ),..., w(x m ) and the integers w(x 1), w(x 2),..., w(x m) are not the same, then at least one of the integers (w(x 1 ), w(x 2 ),..., w(x m ), w(x 1), w(x 2),..., w(x m)) does not belong to one of the sets (w(x 1 ), 21
22 w(x 2 ),..., w(x m )), (w(x 1), w(x 2),..., w(x m)). Let w(x l+1 ) be the smallest of these integers which do not belong to one of these sets, and consider the canonical closures H l, H l of the rings k[x 1, X 2,..., X l ], k[x 1, X 2,..., X l ]. Because of the way X j, X j are chosen, it follows that the rings H l mod t w(xl+1), H l mod t w(x l+1 ) are respectively identical to the rings H mod t w(xl+1), H mod t w(x l+1 ). w(x l+1 ) being by definition larger than w(x l+1 ), the ring H l must contain an element of order w(x l+1 ). Now let (Y 1, Y 2,..., Y ν,... ) be a set of elements of H l chosen in accordance with the statement of Lemma 8 and (p 273) let Y 1, Y 2,..., Y ν be those elements of this set whose orders are smaller than ). The rings w(x l+1 H mod t w(x l+1 ), H l mod t w(x l+1), H l mod t w(x l+1 ) k[y 1, Y 2,..., Y ν ] mod t w(x l+1 ) being identical, there exist elements Y 1, Y 2,..., Y ν of H l such that Y j = Y j mod t w(x l+1 ) (j = 1, 2,..., ν). The canonical closure of k[y 1, Y 2,..., Y ν] which is contained in H l cannot contain any element of order w(x l+1 ). Therefore the canonical closure of k[y 1, Y 2,..., Y ν ] which is none other than H l (since the set (Y 1, Y 2,..., Y ν ) contains the set (X 1, X 2,..., X l )) does not contain an element of order w(x l+1 ) (Lemma 7). Therefore w(x l+1 ) is equal to w(x l+1 ) which contradicts the hypothesis. That the numbers w(x 1 ), w(x 2 ),..., w(x n ) constitutes a subset of the characters of H is established as follows: w(x 1 ) being the smallest element of W ( H) we have w(x 1 ) = χ 1. Assume that w(x l ) is the smallest of the numbers w(x 1 ), w(x 2 ),..., w(x n ) 17 which is not a character of H. w(x l ) would then be contained in the canonical closure of the semigroup generated by the elements of W ( H) which are smaller than w(x l ). Now the elements of W ( H) which are smaller than w(x l ) are contained in W (H l 1 ). We then have w(x l ) W (H l 1 ) which contradicts the choices of the X j. In what follows we will call the numbers w(x 1 ) = χ 1, w(x 2 ) = χ 2,..., w(x m ) = χ m the base characters of H. It follows immediately from the definition of a base of H and from Theorem 4 that every system of elements X 1, X 2,..., X m of H 22
23 such that w( X 1 ) = χ 1, w( X 2 ) = χ 2,..., w( X m ) = χ m constitutes a base of H. A set of elements Y 1, Y 2,..., Y ν of H is called a system of generators, if H is the canonical closure of k[y 1 η 1, Y 2 η 2,..., Y ν η ν ] where η 1, η 2,..., η ν denote the constant terms of Y 1, Y 2,..., Y ν. X 1, X 2,..., X m being a base of H, let us consider a set of elements Y 1, Y 2,..., Y m chosen in the following manner: Y 1 = X 1 + X 1 X 1 k Y 2 = X 2 + X 2 X 2 H 1 Y m = X m + X m X m H m 1 where H i denotes the canonical closure of k[x 1, X 2,..., X i ]; the elements Y 1, Y 2, (p 274)..., Y m clearly constitutes a system of generators for H. Conversely every system of generators contains a subset chosen in this manner. In fact Y 1, Y 2,..., Y ν being a system of generators for H, denote by η 1, η 2,..., η ν the constant terms of Y 1, Y 2,..., Y ν. At least one of the integers w(y 1 η 1 ), w(y 2 η 2 ),..., w(y ν η ν ) is then equal to χ 1, let s say w(y 1 η 1 ) = χ 1. We can then set X 1 = Y 1 η 1. Since W (H 1 ) contains all the elements of W ( H) which are smaller than χ 2, we can choose X i H 1 in such a way that we have w(y i X i) χ 2 (i = 2, 3,..., ν). Section 7: At least one of the integers w(y i X i) is equal to χ 2 ; otherwise the canonical closure of k[x 1, Y 2 X 2,..., Y ν X ν] which is by definition is identical to H does not contain any element of order χ 2. Let w(y 2 X 2) = χ 2. We can then set X 2 = Y 2 X 2 and so on. It follows from these considerations that every system of generators of H contains at least m elements, m being the number of the base characters of H; we call this number the dimension of H. H = k + kt 1 + kt 1 T k[t ]T 1 T 2 T N 1 being a canonical ring, the characters, as well as the base characters, for the rings [I ih ] = H h = k + kt h k[t ]T h+1 T h+2 T N 1 are invariants of H. The characters of H h are clearly determined by those of H. But it is not so for the base characters of H h. 23
24 Consider for example the rings H = k + kt 4ν (1 + t) + kt 6ν (1 + t) + kt 7ν (1 + t) + k[t]t 8ν, H = k + kt 4ν + kt 6ν (1 + t) + kt 7ν (1 + t) + k[t]t 8ν. } (ν > 1) It can easily be checked that these two rings are canonical and that their characters which are also those of the semigroup W ( H) = W ( H ) = {0, 4ν, 6ν, 7ν, 8ν + 1, 8ν + 2, 8ν + 3,... } are the same. These characters are clearly 4ν, 6ν, 7ν, 8ν + 1. Let us now construct a base for H: We can clearly set X 1 = t 4ν (1 + t); k[x 1 ] is a canonical ring and the smallest element of W ( H) not contained in W (k[x 1 ]) is 6ν; we can then set X 2 = t 6ν (1 + t). The canonical closure of k[x 1, X 2 ] is k[x 1, X 2 ] = k + kt 4ν (1 + t) + kt 6ν (1 + t) + k[t]t 8ν. We can then choose X 3 = t 7ν (1 + t) as the third element of the base of H. The canonical closure of k[x 1, X 2, X 3 ] then being equal to H, the base characters of (p 275) H are 4ν, 6ν, 7ν. In a similar manner, we observe that the elements X 1 = t 4ν, X 2 = t 6ν (1 + t), X 3 = t 7ν (1 + t) constitutes a base for H. The base characters of H and H are then the same. Let us now calculate the base characters of the rings H 1 = k + kt 2ν + kt 3ν + k[t]t 4ν, H 1 = k + kt 2ν (1 + t) + kt 3ν (1 + t) + k[t]t 4ν. A base of H 18 1 is formed by t 2ν, t 3ν, t 4ν+1, while the elements t 2ν (1+t), t 3ν (1+t) form a base of H 1, since the canonical closure of k[t 2ν (1 + t), t 3ν (1 + t)] contains the element ( ) t t 4ν (1 + t) 2 t 2ν 3ν 2 (1 + t) (1 + t) = t 4ν+1 (1 + t) t 2ν (1 + t) whose order is 4ν + 1. The base characters of H 1 are then 2ν, 3ν, 4ν + 1 while those of H 1 are 2ν, 3ν. The base characters of the rings [I ih ] = H h constitute then new invariant elements for H. The following considerations allow us to determine successively the base characters of the H h. Consider an arbitrary element of positive order in H. Let T be 24
25 this element and let (X 1, X 2,..., X m ) be a base of H. Denote by χ i the smallest of the numbers χ 1 = w(x 1 ), χ 2 = w(x 2 ),..., χ m = w(x m ), χ m+1 = 19 such that the canonical closure of k[x 1, X 2,..., X i 1, T ] contains 20 an element of order χ i. The elements T, T X 1, T X 2,..., T X i 1, T X i+1,..., T X m constitute then a base of k + HT which is canonical. In fact k[x 1, X 2,..., X i 1, X i+1,..., X m, T ] being the canonical closure of k[x 1, X 2,..., X i 1, X i+1,..., X m, T ], the canonical closure of k[t, T X 1,..., T X i 1, T X i+1,..., T X m ] clearly contains the ring k + T k[x 1, X 2,..., X i 1, X i+1,..., X m, T ]. As k[x 1, X 2,..., X i 1, X i+1,..., X m, T ] contains an element of order χ i, we have k[x 1, X 2,..., X i 1, X i+1,..., X m, T ] = H. The canonical closure of k[t, T X 1,..., T X i 1, T X i+1,..., T X m ] is then identical to k + T H which it contains; since the ring k[t, T X 1,..., T X i 1, T X i+1,..., T X m ] is itself contained in k + T H. Then to show that the elements T, T X 1,..., T X i 1, T X i+1,..., T X m constitute a base of k + T H, it suffices to show that the canonical closures of the (p 276) rings k[t, T X 1,..., T X j ] (1 j < i 1) k[t, T X 1,..., T X i 1 ] k[t, T X 1,..., T X i 1, T X i+1,..., T X h ] (n > h i + 1) do not contain elements of orders, respectively, w(t X j+1 ), w(t X i+1 ), w(t X h+1 ). 25
26 Now these closures are identical respectively to k + T k[x 1, X 2,..., X j, T ] k + T k[x 1, X 2,..., X i 1, T ] k + T k[x 1, X 2,..., X i 1, X i+1,..., X h, T ], where the overline symbol denote always the canonical closure of the corresponding ring. It then suffices to show that the canonical closures of the rings k[x 1,..., X j, T ], k[x 1,..., X i 1, T ], k[x 1,..., X i 1, X i+1,..., X h, T ] do not contain elements of orders w(x j+1 ), w(x i+1 ), w(x h+1 ), respectively. Now the fact that the canonical closure of k[x 1,..., X j, T ] for j < i 1 does not contain any element of order w(x j+1 ) follows from the definition of i. If the ring k[x 1,..., X i 1, T ] contains an element of order w(x i+1 ) or the ring k[x 1,..., X i 1, X i+1,..., X h, T ] an element of order w(x h+1 ), the canonical closure of one of the rings k[x 1, X 2,..., X i 1, X i+2,..., X m, T ], k[x 1, X 2,..., X i 1, X i+1,..., X h, X h+2,..., X m, T ], for h < m 1, k[x 1, X 2,..., X i 1, X i+1,..., X m 1, T ], for h = m 1, contains a system of elements of orders χ 1, χ 2,..., χ m and as a consequence a base of H. This implies the existence of a system of generators of H containing only m 1 elements, contrary to what has been established above (see Section 6). The base characters of k + T H are then w(t ), w(t )+ χ 1, w(t )+ χ 2,..., w(t )+ χ i 1, w(t )+ χ i+1,..., w(t )+ χ m. As the base characters of k + T H do not depend on the choice of the elements X 1, X 2,..., X m, the numbers χ i depend only on T and H. We are going to denote them by χ i = χ(t, H). In a similar manner the characters of k + T H are obtained from those of H (p 277) by the expressions w(t ), χ 1 + w(t ), χ 2 + w(t ),..., χ l + w(t ), for w(t ) χ 1, χ 2,..., χ l, 26
27 and w(t ), χ 1 +w(t ),..., χ j 1 +w(t ), χ j+1 +w(t ),..., χ l +w(t ), for w(t ) = χ j, where we denote the characters of H by χ 1, χ 2, χ l. In particular in the case when all the characters of H are also its base characters, all the characters of k+t H are also its base characters if w(t ) is a character of H or if χ(t, H) is infinite. Remark. ρ being an arbitrary element of W ( H), we can always choose an element T of order w(t ) = ρ of H, in such a way that χ(t, H) is equal to one of the numbers χ 1, χ 2,..., χ m, χ m+1 = which exceeds ρ, provided that ρ is different from the numbers χ i. Suppose in fact that ρ is distinct from the numbers χ 1 < χ 2 < < χ m and let χ l be such that we have χ l < ρ < χ l+1. If X 1, X 2,..., X m is a base of H, the canonical closure of k[x 1, X 2,..., X l ] contains, by definition elements of orders ρ. Let T be one of these elements, and set T = T + X h (with h > l, X m+1 = 0). For l j < h 1 the sets k[x 1, X 2,..., X j, T ] mod t χ h, k[x 1, X 2,..., X j, T ] mod t χ h, k[x 1, X 2,..., X j ] mod t χ h being identical, the ring k[x 1, X 2,..., X j, T ] does not have elements of order w(x j+1 ) = χ j+1. For j < l, ρ = w(t ) being greater than χ j+1, the sets k[x 1, X 2,..., X j ] mod t χ j+1 +1, k[x 1, X 2,..., X j, T ] mod t χ j+1 +1 are identical and consequently k[x 1, X 2,..., X j, T ] does not contain elements of order w(x j+1 ). However the ring k[x 1, X 2,..., X h 1, T ], which contains the element T, contains also the element T T = X h. We then have χ(t, H) = χ h. Let us now consider a canonical semigroup G = G 0 = {0, ν 1, ν 1 + ν 2,..., ν 1 + ν ν N 1 + Nν} (ν N 1 ν). The semigroup G N 1 = Nν 27
28 clearly has only one character which is χ (N 1) 1 = ν. The characters of are then, after the rule indicated above, G N 2 = {0, ν N 1 + Nν} χ (N 2) 1 = ν N 1, χ (N 2) 2 = ν N 1 + ν. The characters of G N 3 are obtained from the previous ones according to the (p 278) same rule: } χ (N 3) 1 = ν N 2, χ (N 3) 2 = ν N 2 + ν N 1, χ (N 3) 3 = ν N 2 + ν N 1 + ν, for ν N 2 > ν N 1 + ν, χ (N 3) 1 = ν N 2, χ (N 3) 2 = ν N 2 + ν N 1, for ν N 2 = ν N 1 + ν χ (N 3) 1 = ν N 2, χ (N 3) 2 = ν N 2 + ν N 1 + ν, for ν N 2 = ν N 1. We obtain successively, by applying always the same rule, the characters χ (i) 1, χ (i) 2,..., χ (i) l i of all the semigroups G i = {0, ν i+1 + G i+1 }. Now let l N 1 = 1, l N 2 = 2, and in general χ (N 1) 1 = ν; χ (N 2) 1 = ν N 1, χ (N 2) 2 = ν N 1 + ν; l i 1 = l i, l i 1 = l i + 1, χ (i 1) 1 = ν i, χ (i 1) 2 = ν i + χ (i) 1,..., χ (i 1) h i = ν i + χ (i) 1, χ (i 1) +1 = ν i + χ (i) +1,..., χ (i 1) l i 1 = ν i + χ (i) l i, for l i, χ (i 1) 1 = ν i, χ (i 1) 2 = ν i + χ (i) 1,..., χ (i 1) h = ν i + χ (i) h 1, χ (i 1) h+1 = ν i + χ (i) h,..., χ (i 1) l i 1 = ν i + χ (i) l i, for = l i + 1, where is any of the positive integers h l i + 1 for which we have ν i < χ (i) h with χ (i) l i +1 =, if ν i χ (i) 1,..., χ (i) l i ; if not χ (i) is the one among χ (i) 1, χ (i) 2,..., χ (i) l i which is equal to ν i. 28
29 It follows immediately from the preceding remarks and the considerations before them that we can always choose the elements T i H i in such a manner that the characters and the base characters of the rings are respectively H N 1 = k[t ], w(t ) = ν, H N 2 = k + H N 1 T N 1, w(t N 1 ) = ν N 1 H i 1 = k + H i T i, w(t i ) = ν i, H = H 0 = k + H 1 T 1, w(t 1 ) = ν 1 The characters χ (N 1) 1 ; The base characters χ (N 1) 1 ; χ (N 2) 1, χ (N 2) 2 ; χ (N 2) 1 χ (N 2) 2 ; χ (i 1) 1, χ (i 1) 2,..., χ (i 1) l i 1 ; χ (0) 1, χ (0) 2,..., χ (0) l 0 ; χ (i 1) 1, χ (i 1) 2,..., χ (i 1) l i 1 ; χ (0) 1, χ (0) 2,..., χ (0) l 0. In particular the base characters of H = H 0 coincide with its characters if (p 279) and only if we choose = l i + 1 every time we had to make a choice; the dimension of H will then be the greatest of the dimensions of the canonical rings having the same characters. Theorem 5. If the base characters χ (N 1) 1 ; χ (N 2) 1, χ (N 2) 2 ;... ; χ (i 1) 1, χ (i 1) 2,..., χ (i 1) li 1,... ; χ (0) 1,..., χ (0) l0 are constructed by setting χ (j) hj = the smallest of the numbers χ (j) 1, χ (j) 2,..., χ (j) lj +1 which are not less than ν j, the dimension of the ring corresponding to H is the smallest possible among the dimensions of canonical rings having the same characters. 29
30 Proof. Let χ (N 1) 1 ; χ (N 2) 1, χ (N 2) 2 ;... ; χ (i 1) 1, χ (i 1) 2,..., χ (i 1) li 1 ;... be another system of base characters, obtained from the same numbers ν j. We have to show that we have l i l i (i = N 1, N 2,..., 0). ν being an arbitrary integer, denote by l i (ν) the number of those χ (i) 1, χ (i) 2,..., χ (i) l i which are not smaller than ν. Similarly let l i (ν) be the number of those χ (i) 1, χ (i) 2,..., χ (i) l i which are not smaller than ν. We will prove, at the same time, that we have l i (ν) l i (ν) l i l i. The equalities l N 1 = l N 1 = 1, l N 2 = l N 2 = 2, l N 1 l N 1 = l N 1 (ν) l N 1 (ν) = 0, being obvious, it suffices to conclude from l N 2 l N 2 = l N 2 (ν) l N 2 (ν) = 0 l i l i, l i (ν) l i (ν) l i l i the inequalities l i 1 l i 1, l i 1 (ν) l i 1 (ν) l i 1 l i 1. We distinguish the following cases: (1) l i = l i, (2) l i l i, (3) l i l i, (4) l i > l i, (5) l i > l i, χ (i) is finite; χ (i) is infinite, χ (i) is infinite, χ (i) is finite, χ (i) is finite, χ (i) is finite; χ (i) is infinite; χ (i) is infinite; χ (i) is finite.; (1) χ (i) being finite, l i (ν i ) is not zero. l i (ν i ) l i (ν i ) being less than or equal (p 280) 30
31 to l i l i = 0 the number l i (ν i ) is not zero. Then χ (i) is finite. It follows that we have l i 1 = l i = l i = l i 1. Let us show that we still have l i 1 (ν) l i 1 (ν) l i 1 l i 1 (= 0) for all ν. According to the recursive formulas χ (i 1) 1 = ν i, χ (i 1) 2 = ν i + χ (i) 1,..., χ (i 1) = ν i + χ (i) 1, χ (i 1) +1 = ν i + χ (i) +1,..., χ (i 1) l i 1 = ν i + χ (i) l i, χ (i 1) 1 = ν i, χ (i 1) 2 = ν i + χ (i) 1,..., χ (i 1) it is clear that we have = ν i + χ (i) 1, χ (i 1) +1 = ν i + χ (i) +1,..., χ (i 1) l i 1 = ν i + χ (i) l i ; l i 1 (ν) = l i, for ν ν i, l i 1 (ν) = l i (ν ν i ) 1, for ν i < ν ν i + χ (i), l i 1 (ν) = l i (ν ν i ), for ν i + χ (i) < ν, l i 1 (ν) = l i, for ν ν i, l i 1 (ν) = l i (ν ν i ) 1, for ν i < ν ν i + χ (i), l i 1 (ν) = l i (ν ν i ), for ν i + χ (i) < ν. It follows that, for ν ν i + min( χ (i), χ (i) ) and for ν > ν 1 + max( χ (i), χ (i) ), we have l i 1 (ν) l i 1 (ν) = l i (ν ν i ) l i (ν ν i ) 0. If χ (i) < χ (i), we have min( χ (i), χ (i) ) = χ (i), max( χ (i), χ (i) ) = χ (i) and l i 1 (ν) l i 1 (ν) = l i (ν ν i ) l i (ν ν i ) 1 < 0 (for ν i + χ (i) < ν ν i + χ (i) ). 31
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