Factorial and Noetherian Subrings of Power Series Rings
aa r X i v : . [ m a t h . AG ] O c t Factorial and Noetherian Subrings of Power Series Rings
Damek Davis and Daqing Wan
Department of MathematicsUniversity of CaliforniaIrvine, CA [email protected]@math.uci.edu
October 29, 2018
Abstract
Let F be a field. We show that certain subrings contained between the polynomialring F [ X ] = F [ X , · · · , X n ] and the power series ring F [ X ][[ Y ]] = F [ X , · · · , X n ][[ Y ]]have Weierstrass Factorization, which allows us to deduce both unique factorization andthe Noetherian property. These intermediate subrings are obtained from elements of F [ X ][[ Y ]] by bounding their total X -degree above by a positive real-valued monotonicup function λ on their Y -degree. These rings arise naturally in studying p -adic analyticvariation of zeta functions over finite fields. Future research into this area may studymore complicated subrings in which Y = ( Y , · · · , Y m ) has more than one variable, andfor which there are multiple degree functions, λ , · · · , λ m . Another direction of studywould be to generalize these results to k -affinoid algebras. Let R be a commutative ring with unity, and let S k be the set of polynomials in R [ X, Y ] = R [ X , · · · , X n ][ Y , · · · , Y m ] that are homogeneous in Y of degree k . Every element of R [ X ][[ Y ]]can be written uniquely in the form f = ∞ X k =0 f k ( X, Y ) , (1)where f k ( X, Y ) is an element of S k . In this expansion, there is no restriction on deg X f k ( X, Y ).Motivated by several applications to the p -adic theory of zeta functions over finite fields, wewant to consider subrings of R [ X ][ Y ]] in which deg X ( f k ) is bounded above by some function λ . In particular, let λ : R ≥ → R ≥ be a monotonic up function. We call λ a growthfunction. Following Wan [9], we define a subring of R [ X ][[ Y ]] as follows: R [ X ; Y, λ ] = { f = ∞ X k =0 f k ( X, Y ) : f k ∈ S k , deg X ( f k ) ≤ C f λ ( k ) , for k ≫ } , (2)1here C f is a constant depending only on f . Since λ is monotonic up, it satisfies the trivialinequality, λ ( x ) + λ ( y ) ≤ λ ( x + y )for all x and y in R ≥ . From this inequality, it is clear that R [ X ; Y, λ ] is an R [ X ]-algebra,which contains R [ X ].If λ is invertible, we have the following equivalent definition: R [ X ; Y, λ ] = { g = ∞ X d =0 g d ( X, Y ) : g d ∈ A d , ord Y ( g d ) ≥ λ − ( C g d ) , for d ≫ } , (3)where A d is the subset of elements of R [[ Y ]][ X ], which are homogeneous of degree d in X ,and ord Y ( g d ) is the largest integer k for which g d is an element of Y k R [ X ][[ Y ]].It is clear that for any positive constant c > R [ X ; Y, cλ ] = R [ X ; Y, λ ]. If λ ( x ) is apositive constant, then R [ X ; Y, λ ] = R [[ Y ]][ X ]. If λ ( x ) = x for all x in R ≥ , then R [ X ; Y, λ ]is called the over-convergent subring of R [ X ][[ Y ]], which is the starting point of Dwork’s p -adic theory for zeta functions. In both of these cases, if R is noetherian, it is known that R [ X ; Y, λ ] is noetherian: when λ is constant, the result follows from Hilbert’s Basis Theorem;the case in which λ ( x ) = x is proved in Fulton [4]. More generally, if R is noetherian and λ satisfies the following inequality, λ ( x ) + λ ( y ) ≤ λ ( x + y ) ≤ λ ( x ) µ ( y )for all sufficiently large x and y , where µ is another positive valued function such that µ ( x ) ≥ x in R ≥ , then R [ X ; Y, λ ] is also noetherian as shown in Wan [9]. Forexample, any exponential function λ ( x ) satisfies the above inequalities. In this case the ringis particularly interesting because it arises naturally from the study of unit root F-crystalsfrom geometry, see Dwork-Sperber [3] and Wan [10] for further discussions.The first condition, λ ( x ) + λ ( y ) ≤ λ ( x + y ), is a natural assumption because it ensuresthat elements of the form (1 − XY ) are invertible, a vital condition to this paper. If λ doesnot grow at least as fast as linear, then (1 − XY ) − = 1 + P ∞ i =1 X k Y k is not an element of R [ X ; Y, λ ]. It is not clear, however, if the second condition, λ ( x + y ) ≤ λ ( x ) µ ( y ), can bedropped. In fact, we have the following open question from Wan [9]. Question 1.1.
Let R be a noetherian ring. Let λ ( x ) be a growth function satisfying λ ( x ) + λ ( y ) ≤ λ ( x + y ) . Is the intermediate ring R [ X ; Y, λ ] always noetherian? This question is solved affirmatively in this paper if R is a field and there is only one Y variable.Throughout this paper we assume that R = F is a field, and that λ grows at least asfast as linear, i.e. λ ( x ) + λ ( y ) ≤ λ ( x + y ) for all x, y ≥
0. Further, we assume that λ (0) = 0and λ ( ∞ ) = ∞ , because normalizing λ this way does not change F [ X ; Y, λ ]. Without lossof generality we also assume that λ is strictly increasing. Finally, we assume that F [ X ; Y, λ ]has only one Y variable. We call an element g = ∞ X d =0 g d ( X , · · · , X n − , Y ) X dn F [ X ; Y, λ ] X n -distinguished of degree s if g s is a unit in F [ X , · · · , X n − ; Y, λ ], andord Y ( g d ) ≥ d > s . The main result of this paper is the following Theorem 1.2.
Under the above assumptions, we have1. (Euclidean Algorithm) Suppose that g is X n -distinguished of degree s in F [ X ; Y, λ ] , andthat f is an element of F [ X ; Y, λ ] . Then there exist unique elements, q in F [ X ; Y, λ ] ,and r in the polynomial ring F [ X , · · · , X n − ; Y, λ ][ X n ] with deg X n ( r ) < s , such that f = qg + r .2. (Weierstrass Factorization) Let g be X n -distinguished of degree s . Then there existsa unique monic polynomial ω in F [ X , · · · , X n − ; Y, λ ][ X n ] of degree s in X n and aunique unit e in F [ X ; Y, λ ] such that g = e · ω . Further, ω is distinguished of degree s .3. (Automorphism Theorem) Let g ( X, Y ) = P µ g µ ( Y ) X µ be an element of F [ X ; Y, λ ] where µ = ( µ , · · · , µ n ) and X µ = X µ · · · X µ n n . If g µ ( Y ) is not divisible by Y for some µ , where µ n > , then there exists an automorphism σ of F [ X ; Y, λ ] such that σ ( g ) is X n -distinguished.4. F [ X ; Y, λ ] is noetherian and factorial. The Euclidean algorithm is the key part of this theorem. Our proof of this algorithmfollows Manin’s proof of the analogous result for power series rings as written in Lang [7],except that we have to keep careful track of more delicate estimates that arise from thegeneral growth function λ ( x ). The other results are classical consequences of this algorithm,which are proved in this paper, but the techniques are essentially unchanged from techniquesutilized in proofs of analogous results for power series rings as given in Bosch, etc. [1].This topic is also motivated by a considerable body of work concerning “ k -affinoid”algebras from non-Archimedean analysis. Let k be a complete non-Archimedean valuedfield, with a non-trivial valuation, and define T n = k h X , · · · , X n i , Tate’s algebra, to be thealgebra of strictly convergent power series over k : T n = { P µ a µ X µ : | a µ | | µ |→∞ → } . Thealgebra, T n , is a noetherian and factorial ring with many useful properties, and it is the basisfor studying k -affinoid algebras, see Bosch etc [1]. A k -algebra, A , is called k -affinoid if thereexists a continuous epimorphism, T n → A , for some n ≥
0. Given ρ = ( ρ , · · · , ρ n ) in R n ,where ρ i > i , one can define T n ( ρ ) = { X µ a µ X µ ∈ k [[ X , · · · , X n ]] : | a µ | ρ µ · · · ρ µ n n | µ |→∞ → } . Note that T n (1 , · · · ,
1) = T n . Furthermore, T n ( ρ ) is k -affinoid if, and only if, ρ i is an elementof | k ∗ a | for all i , where k a is the algebraic closure of k , from which one can immediately verifythat it is noetherian. It is shown by van der Put in [8] that this ring is noetherian for any ρ in R n , where ρ i > i . Define the Washnitzer algebra W n to be W n = [ ρ ∈ R n ,ρ i > T n ( ρ ) .
3t is shown in G¨untzer [5] that W n is noetherian and factorial. A motivating study of W n isgiven by Grosse-K¨onne [6]. This overconvergent ring W n is also the basis (or starting point)of the Monsky-Washnitzer formal cohomology and the rigid cohomology.More generally, for a growth function λ ( x ), we can also define T n ( ρ, λ ) = { X µ a µ X µ ∈ k [[ X , · · · , X n ]] : | a µ | ρ λ − ( µ )1 · · · ρ λ − ( µ n ) n | µ |→∞ → } . Similarly, define W n ( λ ) = [ ρ ∈ R n ,ρ i > T n ( ρ, λ ) . If λ is invertible, W n ( λ ) most closely resembles the ring k [ X ; Y, λ ] = k [ X , · · · , X n ; Y, λ ]studied in this paper. If λ ( x ) = cx for some c > x in R ≥ , then W n ( λ ) = W n is theWashnitzer algebra. Similarly, T n ((1 , · · · , , λ ) = T n for all λ , and T n ( ρ, id) = T n ( ρ ).The results of this paper suggest that there may be a p -adic cohomology theory for moregeneral growth functions λ ( x ) (other than linear functions), which would help to explainthe principal zeroes of Dwork’s unit root zeta function [2], [10] in the case when λ is theexponential function. This is one of the main motivations for the present paper. Acknowledgments . We would like to thank Christopher Davis for informing us ofseveral relevant references.
For the rest of the paper, we assume that F is a field and that p is a fixed positive realnumber greater than one. Definition 2.1.
Define | | λ on F [[ Y ]] by | f ( Y ) | λ = p λ ( ord Y ( f )) for all f in F [[ Y ]] . This is basically the Y -adic absolute value on F [[ Y ]], re-scaled by the growth function λ ( x ). Proposition 2.2. ( F [[ Y ]] , | | λ ) is a complete normed ring.Proof. We defined λ (0) = 0 and λ ( ∞ ) = ∞ , so | a | λ = 0 if, and only if, a = 0 (because λ isstrictly increasing), and | c | = 1 for all c in F × . Suppose that f and g are elements of F [[ Y ]],then ord Y ( f g ) = ord Y ( f ) + ord Y ( g ), and λ (ord Y ( f ) + ord Y ( g )) ≥ λ (ord Y ( f )) + λ (ord Y ( g )).Thus, | f g | λ = 1 p λ (ord Y ( f )+ord Y ( g )) ≤ p λ (ord Y ( f ))+ λ (ord Y ( g )) = | f | λ | g | λ . Y ( f + g ) ≥ min { ord Y ( f ) , ord Y ( g ) }| f + g | λ = 1 p λ (ord Y ( f + g )) ≤ p λ (min { ord Y ( f ) , ord Y ( g ) } ) = max {| f | λ , | g | λ } . To show completeness, if (cid:0) f ( i ) (cid:1) ∞ i =1 is a Cauchy sequence with respect to the standard Y -adic norm | | on F [[ Y ]], then f ( i ) converges to an element f in F [[ Y ]]. Thus, | f − f ( i ) | = p ord Y ( f − f ( i )) converges to 0 as i approaches ∞ , and so ord Y ( f − f ( i ) ) must approach ∞ . Thiscan happen if, and only if, the corresponding sequence, λ (ord Y ( f − f ( i ) )), approaches ∞ , asdesired. Remark 2.3.
A norm which only satisfies | ab | ≤ | a || b | , instead of strict equality is sometimescalled a pseudo-norm; we disregard the distinction in this paper. We can write any element f in F [ X ; Y, λ ] in the following form: f ( X, Y ) = X µ f µ ( Y ) X µ where µ = ( µ , · · · , µ n ) is a tuple of positive integers, and X µ = X µ · · · X µ n n . This form andthe above norms allow us to formulate two equivalent definitions for F [ X ; Y, λ ]: F [ X ; Y, λ ] = { f = ∞ X µ f µ ( Y ) X µ : f µ ∈ F [[ Y ]] , | f µ | p λ − ( C f | µ | ) | µ |→∞ → } = { f = ∞ X µ f µ ( Y ) X µ : f µ ∈ F [[ Y ]] , | f µ | λ p C f | µ | | µ |→∞ → } where | µ | = µ + · · · + µ n . Definition 2.4.
For all c in R n , c i > , define F [ X ; Y, λ ] c = { X ν f µ X µ : | f µ | λ p c · µ | µ |→∞ → } where c · µ = c µ + · · · + c n µ n . Definition 2.5.
Define k k λ,c on F [ X ; Y, λ ] c by k f k λ,c = max | f µ | λ p c · µ . It’s easy to see that F [[ Y ]] ⊂ F [ X ; Y, λ ] c ⊆ F [ X ; Y, λ ] c ′ if c ′ i ≤ c i for i = 1 , · · · , n . Proposition 2.6.
The function k k λ,c is a non-Archimedean norm on F [ X ; Y, λ ] c . roof. On F [[ Y ]], k k λ,c reduces to | | λ . Suppose that f and g are elements of F [ X ; Y, λ ] c .Then k f + g k λ,c = max µ {| f µ + g µ | λ p c · µ }≤ max µ { max {| f µ | λ , | g µ | λ } p c · µ }≤ max {k f k λ,c , k g k λ,c } . Next, k f g k λ,c = max σ { (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X µ + ν = σ f µ g ν !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ p c · σ }≤ max σ { max µ + ν = σ {| f µ g ν | λ } p c · σ }≤ max µ,ν {| f µ | λ | g ν | λ p c · ( µ + ν ) } = k f k λ,c k g k λ,c . Proposition 2.7. F [ X ; Y, λ ] = [ c ∈ R n ,c i > F [ X ; Y, λ ] c Proof.
Suppose that f is an element of F [ X ; Y, λ ], then | f µ | λ p ( C f , ··· ,C f ) · µ converges to 0 as | µ | approaches ∞ . Conversely, if f is an element of F [ X ; Y, λ ] c , let C f = min i c i . Lemma 2.8.
Suppose f is an element of F [ X ; Y, λ ] . Then f ( X, Y ) is invertible if, and onlyif, f ≡ c mod ( Y ) where c is a unit in F . If an element f in F [ X ; Y, λ ] c is invertible in F [ X ; Y, λ ] , then f − is an element of F [ X ; Y, λ ] c . Further, if k f k λ,c ≤ , then k f − k λ,c ≤ .Proof. If f ( X, Y ) is invertible, then it is an invertible polynomial modulo ( Y ). Therefore, f is a non-zero unit modulo ( Y ).If f ≡ c mod ( Y ) for c in F × , we can write f = c (1 − g ( X, Y )) as an element of F [ X ; Y, λ ] c for some c >
0. Then f − = c − ∞ X k =1 g ( X, Y ) k ! = c − ∞ X k =1 ∞ X j =1 X µ (1) + ··· + µ ( k ) = σ | σ | = j k Y i =1 g µ ( i ) ( Y ) X σ Observe that, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 g µ ( i ) ( Y ) X σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ! p c · σ = k Y i =1 (cid:16)(cid:12)(cid:12) g µ ( i ) ( Y ) X σ (cid:12)(cid:12) λ p c · µ ( i ) (cid:17) | σ | approaches ∞ because g µ = − c f µ . Suppose k f k λ,c ≤
1, then thisproduct is also less than or equal to one, because each term satisfies this property, so k f − k λ,c = | c − | λ = 1. Proposition 2.9.
The ring, ( F [ X ; Y, λ ] c , k k λ,c ) , is an F [[ Y ]] -Banach algebra.Proof. Suppose that f = P µ f µ ( Y ) X µ and g = P ν g ν ( Y ) X ν , are elements of F [ X ; Y, λ ] c ,then | f µ ± g µ | λ ≤ max {| f µ | λ , | g µ | λ } , and the quantity max {| f µ | λ p c · µ , | g µ | λ p c · µ } converges to 0as | µ | approaches ∞ . Thus, f + g is an element of F [ X ; Y, λ ] c .Similarly, we see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X µ + ν = σ f µ g ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ≤ max µ + ν = σ {| f µ | λ · | g ν | λ } and lim | σ |→∞ max µ + ν = σ {| f µ | · | g ν | p c · σ } = 0 as desired. Thus, f g is an element of F [ X ; Y, λ ] c .Now to prove that that this norm is complete, we let (cid:0) f ( i ) (cid:1) ∞ i =1 = (cid:16)P ∞ µ f ( i ) µ X µ (cid:17) ∞ i =1 bea Cauchy sequence in F [ X ; Y, λ ] c . Then we can choose a suitable subsequence of (cid:0) f ( i ) (cid:1) ∞ i =1 (because a Cauchy sequence is convergent if, and only if, it has a convergent subsequence)and assume that | f ( j ) µ − f ( i ) µ | λ p c · µ ≤ k f ( j ) − f ( i ) k λ,c < /i for all j > i > | µ | ≥ . For all j and µ , f ( j ) µ is an element of F [[ Y ]] which is complete, so there is an element f µ in F [[ Y ]] such that f ( j ) µ converges to f µ as j approaches ∞ . Define f = P µ f µ X µ . We claimthat | f µ | λ p c · µ converges to 0 as | µ | approaches ∞ .Note that | | λ is continuous, so | f µ − f ( i ) µ | λ p c · µ ≤ /i , for all | µ | ≥ i >
0. Wechoose µ , such that | µ | is sufficiently large, so that | f ( i ) µ | λ p c · µ < /i . Since the norm is non-Archimedean, this shows that | f µ | λ p c · µ ≤ /i . Thus, | f µ | λ p c · µ converges to 0 as | µ | approaches ∞ . Hence, f is an element of F [ X ; Y, λ ] c and k f − f ( i ) k λ,c = max | f µ − f ( i ) µ | λ p c · µ ≤ /i ,therefore, lim i f ( i ) = f . Definition 2.10.
A power series f ( X, Y ) = P ∞ k =0 f k ( X, Y ) X kn in F [ X ; Y, λ ] is called X n -distinguished of degree s in F [ X ; Y, λ ] if1. f s ( X, Y ) is a unit in F [ X , · · · , X n − ; Y, λ ] and2. | f k ( X, Y ) | < for all k > s .Equivalently, f mod ( Y ) is a unitary polynomial in X n of degree s .A power series f ( X, Y ) = P ∞ k =0 f k ( X, Y ) X kn in F [ X ; Y, λ ] c is called X n -distinguished ofdegree s in F [ X ; Y, λ ] c if1. f s ( X, Y ) is a unit in F [ X , · · · , X n − ; Y, λ ] c and k f s ( X, Y ) k λ, ( c , ··· ,c n − ) = 1 ,2. k f k λ,c = k f s ( X, Y ) X sn k λ,c = p c n s > k f k ( X, Y ) X kn k λ,c for all k = s .
7f an element f in F [ X ; Y, λ ] is X n -distinguished of degree s in F [ X ; Y, λ ], then it is X n -distinguished in F [ X ; Y, λ ] c for some c in R n . Indeed, suppose that f is an element of F [ X ; Y, λ ] c . Since f s ( X, Y ) is a unit, we can write f s ( X, Y ) = u + h , where u is a unit in F [[ Y ]], and h is an element of ( Y ). By choosing c , · · · , c n − small enough, we can make k h k λ,c <
1, and so k f s ( X, Y ) k λ, ( c , ··· ,c n − ) = 1. We can reduce c , · · · , c n − even further toensure that k f k ( X, Y ) X kn k λ,c < k f s ( X, Y ) X sn k λ,c = p c n s , because f k is an element of ( Y ) forall k > s . Now, to ensure that k f k ( X, Y ) X kn k λ,c < k f s ( X, Y ) X sn k λ,c = p c n s , for k < s , we canshrink c , · · · , c n − once again so that k f k ( X, Y ) k λ,c < p c n . In this way we find that for all k < s , k f k ( X, Y ) X kn k λ,c < p c n p c n ( s − = p c n s = k f s ( X, Y ) X sn k λ,c as desired.We can use the notion of X n -distinguished elements to derive a Euclidean algorithmfor F [ X ; Y, λ ] c . This Euclidean algorithm will then produce Weierstrass factorization for X n -distinguished elements, which will allow us to deduce that F [ X ; Y, λ ] is noetherian andfactorial.
Theorem 2.11.
Let g = ∞ X k =0 g k ( X, Y ) X kn be X n -distinguished of degree s in F [ X ; Y, λ ] c . Then every f in F [ X ; Y, λ ] c can be writtenuniquely in the form f = qg + r where q is and element of F [ X ; Y, λ ] c and r is a polynomial in F [ X , · · · , X n − ; Y, λ ] c [ X n ] ,with deg X n ( r ) < s . Further, if f and g are polynomials in X n , then so are q and r .Proof. Let α , τ be projections given by, α : ∞ X k =0 g k ( X, Y ) X kn s − X k =0 g k ( X, Y ) X kn τ : ∞ X k =0 g k ( X, Y ) X kn ∞ X k = s g k ( X, Y ) X k − sn We see that τ ( w ) and α ( w ) are elements of F [ X ; Y, λ ] c , and that τ ( wX sn ) = w . It is alsoclear that τ ( w ) = 0 if, and only if, deg X n ( w ) < s , for all w in F [ X ; Y, λ ] c .Such q and r exist if, and only if, τ ( f ) = τ ( qg ). Thus, we must solve, τ ( f ) = τ ( qα ( g )) + τ ( qτ ( g ) X sn ) = τ ( qα ( g )) + qτ ( g ) . Note that τ ( g ) is invertible, trivially, because it is congruent to a unit modulo ( Y ). Let M = qτ ( g ). Thus, we can write τ ( f ) = τ (cid:18) M α ( g ) τ ( g ) (cid:19) + M = (cid:18) I + τ ◦ α ( g ) τ ( g ) (cid:19) M.
8e want to show that the map (cid:16) I + τ ◦ α ( g ) τ ( g ) (cid:17) − exists.Suppose z is an element of F [ X ; Y, λ ] c . We first claim that k τ ( z ) k λ,c ≤ k z k λ,c p cns . Indeed,there exists µ such that k z k λ,c = | z µ | λ p c · µ ≥ | z ν | λ p c · ν for all ν Thus, k z k λ,c p c n s = | z µ | λ p c · µ − c n s ≥ | z ν | λ p c · ν − c n s for all ν. The maximum over all ν with ν n ≥ s is equal to k τ ( z ) k λ,c , as asserted. Thus k τ ( g ) k λ,c ≤ p cns p cns = 1, so by the lemma 2.8 k τ ( g ) − k ≤
1. Let h = α ( g ) τ ( g ) . Then, k h k λ,c ≤ k α ( g ) k λ,c k τ ( g ) − k λ,c < p c n s . Next we claim that k ( τ ◦ h ) m ( z ) k λ,c < k z k λ,c k h k mλ,c p mcns , for all m in N . Indeed, k τ ( zh ) k λ,c ≤ k zh k λ,c p cns ≤ k z k λ,c k h k λ,c p cns by what we just proved. Now, assume that this is true for m , then k τ (( τ ◦ h ) m ( z ) h ) k λ,c ≤ k ( τ ◦ h ) m ( z ) k λ,c k h k λ,c p c n s ≤ k z k λ,c k h k mλ,c p mc n s k h k λ,c p c n s = k z k λ,c k h k m +1 λ,c p ( m +1) c n s Now, we know that, ( I + τ ◦ h ) − ( z ) = z + ∞ X m =1 ( − m ( τ ◦ h ) m ( z ) . Let w ( i ) ( z ) = z + P im =1 ( − m ( τ ◦ h ) m ( z ). We claim that the sequence (cid:0) w ( i ) ( z ) (cid:1) ∞ i =1 is Cauchyfor every z in F [ X ; Y, λ ] c . Indeed, k w ( i +1) ( z ) − w ( i ) ( z ) k λ,c = k ( τ ◦ h ) i +1 ( z ) k λ,c ≤ k z k λ,c k h k i +1 λ,c p ( i +1) c n s . Since k h k λ,c < p c n s , we see that this difference approaches 0 as i approaches ∞ . Sincethis norm is non-Archimedean, this is all we need to show. Therefore, since F [ X ; Y, λ ] c iscomplete, we see that w ( z ) = lim i w ( i ) ( z ) exists for every z in F [ X ; Y, λ ] c . Uniqueness isimmediate from the invertibility of the map.To prove the last statement, note that we could already carry out division uniquely inthe ring F [ X , · · · , X n − ; Y, λ ] c [ X n ], by the polynomial Euclidean algorithm. Therefore, thedivision is unique in F [ X ; Y, λ ] c . 9 orollary 2.12. Suppose that g is X n -distinguished of degree s in F [ X ; Y, λ ] , and that f is an element of F [ X ; Y, λ ] . Then there exist unique elements, q in F [ X ; Y, λ ] , and r in thepolynomial ring F [ X , · · · , X n − ; Y, λ ][ X n ] with deg X n ( r ) < s , such that f = qg + r .Proof. Choose c for which f and g are elements of F [ X ; Y, λ ] c . Then, choose c ′ i ≤ c i suchthat g is X n -distinguished of degree s in F [ X ; Y, λ ] c ′ . Carry out the division in this ring.To show uniqueness we observe that | f | = max {| q | , | r |} . Indeed, without loss of gen-erality we can assume that max {| q | , | r |} = 1. Thus, | f | ≤
1. Suppose that | f | < ≡ qg + r mod ( Y ). Since deg X n ( r ) < s = deg X n ( g mod ( Y )), we must have that q = r ≡ Y ), contradicting max {| q | , | r |} = 1. Thus, if q ′ g + r ′ = qg + r , then( q − q ′ ) g + ( r − r ′ ) = 0, thus, | q − q ′ | = | r − r ′ | = 0. Theorem 2.13.
Let f be X n -distinguished of degree s . Then, there exists a unique monicpolynomial ω in F [ X , · · · , X n − ; Y, λ ][ X n ] of degree s in X n and a unique unit e in F [ X ; Y, λ ] such that f = e · ω . Further, ω is distinguished of degree s .Proof. By the previous theorem there exists an element q in F [ X ; Y, λ ] and a polynomial r in F [ X , · · · , X n − ; Y, λ ][ X n ] such that deg X n ( r ) < s and X sn = qf + r . We let ω = X sn − r . ω = qf is clearly X n -distinguished of degree s . Since the X n -degree of ω mod ( Y ) is thesame the X n -degree of f mod ( Y ), we see that q is a unit. Set e = q − , yielding f = e · ω .Uniqueness is immediate from the uniqueness of the division algorithm. Definition 2.14.
A Weierstrass polynomial is a monic X n -distinguished polynomial in F [ X , · · · , X n − ; Y, λ ][ X n ] . Lemma 2.15.
Let σ be the map such that σ ( X i ) = ( X i + X dj ) with d ≥ , and σ ( X j ) = X j for all j = i . Then σ is a well-defined automorphism of F [ X ; Y, λ ] .Proof. Without loss of generality, we can assume that i = n and j = 1. Let f ( X, Y ) = P µ f µ ( Y ) X µ = P µ f µ ( Y ) X µ · · · X µ n n . Observe that, σ ( f ) = X µ f µ ( Y ) X µ · · · X µ n − n − ( X n + X d ) µ n = X µ f µ ( Y ) X µ · · · X µ n − n − µ n X j =0 (cid:18) µ n j (cid:19) X dj X µ n − jn . The quantity dj + µ n − j = ( d − j + µ n is maximal when j = µ n . Therefore, the aboveconverges if X µ f µ ( Y ) X µ + dµ n · · · X µ n − n − X µ n n converges. Choose c so that | f µ ( Y ) | p c · µ converges to 0 as | µ | approaches ∞ . Let c ′ n = c n d +1) .Then, c µ + · · · + c n µ n > c µ + · · · + c ′ n µ n ( d + 1), so | f µ ( Y ) | p c µ + ··· + c ′ n µ n ( d +1) converges to 0as | µ | approaches ∞ . Therefore, this map is well defined with inverse, σ − ( X n ) = X n − X d and σ − ( X j ) = X j , if j = n . 10 heorem 2.16. Suppose f ( X, Y ) = P µ f µ ( Y ) X µ is an element of F [ X ; Y, λ ] . If | f µ ( Y ) | =1 , for some µ , where µ n > , then there exists an automorphism σ of F [ X ; Y, λ ] such that σ ( f ) is X n -distinguished.Proof. Let f ( X, Y ) = P µ f µ ( Y ) X µ = P µ f µ ( Y ) X µ · · · X µ n n . Let ν = ( ν , · · · , ν n ) be themaximal n -tuple, with respect to lexicographical ordering, such that f ν ( Y ) is not an elementof ( Y ). Let t ≥ max ≤ i ≤ n µ i for all indices µ such that f µ ( Y ) is not an element of ( Y ), e.g.,let t be the total X -degree of f ( X, Y ) mod ( Y ). Now, define an automorphism σ ( X i ) = X i + X d i n for i = 1 , · · · , n −
1, and σ ( X n ) = X n , where d n = 1, and d n − j = 1 + t P j − k =0 d n − k ,for j = 1 , · · · , n −
1. We see that this map is just a finite composition of automorphisms ofthe same type as given above. Hence, it is an automorphism.We will prove that σ ( f ) is X n -distinguished of degree s = P ni =1 d i ν i . First, for all µ suchthat f µ ( Y ) is a unit, and µ = ν , we have P ni =1 d i µ i < s : There exists an index q such that1 ≤ q ≤ n , such that µ = ν , · · · , µ q − = ν q − and µ q < ν q . Therefore µ q ≤ ν q − n X i =1 d i µ i ≤ q − X i =1 d i ν i + d q ( ν q −
1) + t n X i = q +1 d i = q X i =1 d i ν i − < n X i =1 d i ν i = s. Now, σ ( f ) = X µ f µ ( Y )( X + X d n ) µ · · · ( X n − + X d n − n ) µ n − X µ n n ≡ X µf µ ( Y ) / ∈ ( Y ) f µ ( Y ) X λ , ··· ,λ n − ≤ λ i ≤ µ i (cid:18) µ λ (cid:19) · · · (cid:18) µ n − λ n − (cid:19) X µ − λ · · · X µ n − − λ n − n − X d λ + ··· + d n − λ n − + µ n n ≡ X g i X in mod ( Y )where the g i are elements of F [ X , · · · , X n − ]. Therefore, σ ( f ) mod ( Y ) is a polynomial in X n of degree less than or equal to s , and X d λ + ··· + d n − λ n − + µ n n = X sn if, and only if, µ n = ν n and λ i = µ i = ν i for i = 1 , · · · , n −
1. Thus, we have g s = f ν ( Y ) mod ( Y ), but f ν ( Y ) isnot an element of ( Y ), and so σ ( f ) is a unitary polynomial modulo ( Y ). Therefore, σ ( f ) is X n -distinguished of degree s . Theorem 2.17.
Let ω be a Weierstrass polynomial of degree s in X n . Then for all d ≥ Y d F [ X ; Y, λ ] /Y d ωF [ X ; Y, λ ] is a finite free F [ X , · · · , X n − ; Y, λ ] -module, and2. Y d F [ X , · · · , X n − ; Y, λ ][ X n ] /Y d ωF [ X , · · · , X n − ; Y, λ ][ X n ] is isomorphic to Y d F [ X ; Y, λ ] /Y d ωF [ X ; Y, λ ] .Proof. Suppose that g is an element of Y d F [ X ; Y, λ ], then g = Y d h for some element h in F [ X ; Y, λ ]. Since ω is X n -distinguished, there exists a unique element q in F [ X ; Y, λ ],and a unique polynomial r in F [ X , · · · , X n − ; Y, λ ][ X n ] with deg X n ( r ) < s , such that h = qω + r , so g = qY d ω + Y d r . Therefore, g ≡ Y d r mod Y d ωF [ X ; Y, λ ], so the set { Y d , Y d X n , · · · , Y d X s − n } forms a generating set of Y d F [ X ; Y, λ ] /Y d ωF [ X ; Y, λ ] over thering F [ X , · · · , X n − ; Y, λ ]. The natural map Y d F [ X , · · · , X n − ; Y, λ ][ X n ] → Y d F [ X ; Y, λ ] /Y d ωF [ X ; Y, λ ]is thus surjective. The kernel of this map is Y d ωF [ X , · · · , X n − ; Y, λ ][ X n ], trivially.11 heorem 2.18. F [ X ; Y, λ ] = F [ X , · · · , X n ; Y, λ ] is factorial, for all n ≥ .Proof. First assume that n = 1. Suppose that f is an element of F [ X ; Y, λ ]. Write f = e · Y d ω , where ω is a unitary polynomial in X of degree s in F [[ Y ]][ X ], and e isa unit in F [ X ; Y, λ ]. We can factor ω = uq · · · q m into irreducible factors and a unit in F [[ Y ]][ X ] because this ring is factorial. We want to show that these factors are still irre-ducible in F [ X ; Y, λ ]. Suppose that q i is not irreducible modulo ωF [[ Y ]][ X ], then q i ≡ ab mod ω , so there exists g = 0 such that q i = ab + gω . However, by the uniqueness of thedivision algorithm g = 0, thus, a or b is a unit modulo ω . Therefore, q i is irreducible in F [[ Y ]][ X ] /ωF [[ Y ]][ X ] ≃ F [ X ; Y, λ ] /ωF [ X ; Y, λ ].If q i is not irreducible in F [ X ; Y, λ ], then there exists elements a and b in F [ X ; Y, λ ],such that q i = ab . Without loss of generality, b must be a unit modulo ω , so b = c + gω .Write q i = a ( c + gω ) = ac + agω . However, by the uniqueness of the division algorithm, thesame representation of the division algorithm which holds in F [[ Y ]][ X ], holds in F [ X ; Y, λ ],and since deg X ( ac ) < s , we must have ag = 0. This is a contradiction. Therefore, the q i are irreducible in both rings. Write f = eu · Y d q · · · q m uniquely as a product of irreduciblefactors and a unit. Continue by induction. Theorem 2.19. F [ X , · · · , X n ; Y, λ ] is noetherian.Proof. Assume first that n = 1. Let I ⊆ F [ X ; Y, λ ] be an ideal. Suppose that d is the largestpositive integer such that I ⊆ Y d F [ X ; Y, λ ]. Then every f in I is divisible by Y d . Choosean element f in I such that ord Y f = d . We can then write f = e · Y d ω for some unit e ,and Weierstrass polynomial ω . Consider the image of I in Y d F [ X ; Y, λ ] /Y d ωF [ X ; Y, λ ] ≃ Y d F [[ Y ]][ X ] /Y d ωF [[ Y ]][ X ]; this is Noetherian. Therefore, we can pull back the finite list ofgenerators for the image of I and add Y d ω to get a finite generating system for I . Continueby induction. This paper resolves the open problem left in Wan [9], stated at the beginning of the paper,only when F is a field and when F [ X ; Y, λ ] has only one Y variable. It would be interestingto settle the general case (either positively or negatively) when Y has more than one variableand R is a general noetherian ring.Another open question is whether F [ X ; Y, λ ] is factorial if there is more than one Y variable. The answer to this question cannot be obtained from the same methods usedin this paper because elements exist that cannot be transformed into an X n distinguishedelement through an automorphism. For example: f ( X, Y ) = Y + XY + X Y + X Y + · · · . Another direction of research could involve studying the algebras T n ( ρ, λ ) and W n ( λ ).One could try to generalize results only known about the overconvergent case ( λ ( x ) = id ),such as those proven in Gross-Kl¨onne [6]. One could also try to develop the k -affinoid theoryof T n ( ρ, λ ) and W n ( λ ). 12 eferenceseferences