"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Macauley, Ronald Alvin"@en . "2012-03-10T00:37:51Z"@en . "1951"@en . "Master of Arts - MA"@en . "University of British Columbia"@en . "If R is a field on which all (non-archimedean) valuations are known, then all valuations on R[x], where x is transcendental over R , are also known. Ostrowski described such valuations of R[x] by means of pseudo-convergent sequences in the algebraic completion of A of R . MacLane later showed that if all valuations of R are discrete, then any valuation V of R [x] can be represented by certain \"key\" polynomials in R [x]. The present paper exhibits the connection between these two treatments. This is achieved by first determining keys for the valuation which a pseudo-convergent sequence defines on A[x], and then relating these keys to those for V ."@en . "https://circle.library.ubc.ca/rest/handle/2429/41331?expand=metadata"@en . "VALUATIONS OF POLYNOMIAL RINSS by Ronald Alvin Maeauley Cef*\ A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of MATHEMATICS .We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS. Members of. the Department \u00C2\u00ABf *@thematics -THE UNIVERSITY OF BRITISH COLUMBIA April, 1951 Abstract If R i s a f i e l d on which a l l (nori-archimedean) valua-tions are known, then a l l valuations on Rfx^J, where x- i s transcendental over R , are also known. Ostrowski described such valuations of R[xl by means of pseudo-convergent se-quences i n the algebraic completion o f A of R . MacLane later showed that i f a l l valuations of R are discrete, then any valuation V of R [x\"] can be represented by certain \"key\" polynomials in R [x}. The present paper exhibits the connection between these two treatments. This is achieved by f i r s t determining keys for the valuation which a pseudo-convergent sequence defines xm ATx], and then relating these keys to those for V . A c k n o w l e d g m e n t The writer wishes to express his thanks to Dr. B.N. Moyls of the Department of Mathematics at the University of British ColumMa for his advice and guidance. His numerous criticisms and suggestions proved invaluable i n the preparation of t h i s thesis. 1. Introduction. A non-archimedean valuation V , hereafter simply called a valuation, of an integral domain R i s a single-valued mapping of the elements of R Into the real numbers and + 0 0 such that: 1) Va i s a unique f i n i t e r e a l number for a / 0 , 2) VO = + op ', 3) V(ab) - Va + Vb for a l l a, b e R , 4) V(a + b) > min fVa, Vb} for a l l a, b e R . An extremely Important property of these valuations is that, i f Va ^ Vb , then Via +b) = min {Va, Vb}; hence, i f V(a +b) > Va, Va \u00C2\u00AB\u00C2\u00AB Vb . Ostrowski, and later MacLane, attacked the problem of finding a l l extensions of valuations on an integral domain R to the ring of polynomials R f x], where x i s transcendental over R \u00E2\u0080\u00A2 MacLane*s results are based on the assumption that a l l valuations of R are discrete; that i s , the real numbers used as values form an isolated point set. It i s the purpose of this paper to provide a connection between the valuations of Ostrowski and MacLane on R fx] , where R is a f i e l d with only discrete valuations. Definition 1 .1: Let R be a f i e l d with a valuation V . The sequence {a^J, where e R , is a pseudo-convergent sequence with respect to V i f V(a^ - a^^) < V ( a i + 1 - c^) for a l l i > N , so B E fixed positive integer. If {a.} i s a pseudo-convergent sequence with respect to 2 V , then the sequence /Vc^ } is eventually strictly monotone increasing or eventually attains a constant value; as long as 0 is not a limit of { Va . Definition 1.3: For a, b e K, a.1 equivalence divides b in V i f there exists c e K such that b <~ ea (V) ; notation: 3. a|b (V) . If V is any valuation of R[x] which reduces to a discrete valuation V 0 of R , MacLane ([2]) represents V by the following inductive method: a value Vjx = Vx = (i^ is assigned to x \u00E2\u0080\u00A2 Then f o r any polynomial f(x) \u00C2\u00A3 R[x], f (x) - a nx n + a ^ j x 1 1 \" 1 + ... + a Q, a function Vi on R\u00C2\u00A3x] i s defined by V-jf(x) \u00C2\u00BB m i n f v ^ i + i ^ i j \u00E2\u0080\u00A2 This function may be shown to be a valuation of R fx} such that < V ; that i s , Vig(x) < vg(x) for a l l g(x) e R[x]. The value i s called a f i r s t stage value and i s symbolized by = \u00C2\u00A3 v o , V]x - \u00E2\u0080\u00A2 Either V x V , that i s , V xg(x) \u00C2\u00BB Vg(x) for a l l g(x) e R[x}, or there exists an f (x) e R[x] such that Vjf (x) < Vf (x) . If that latter i s the case, 92(x) e a [ x l i s chosen such that 92 i s a monic polynomial of. the smallest degree satisfying V^92 < 2 \u00E2\u0080\u00A2 This polynomial satisfies, over 1^ , MacLane's con-ditions for a key polynomial. Definition 1.4: I*et W be any valuation of Rfx] \u00E2\u0080\u00A2 A poly-nomial 9 e Rfx] i s a key polynomial over .the value W i f : (i) 9 i s equivalence irreducible - 9|a(x)b(x) (W) implies either 9|a(x) (W) or 9|b(x) (W) , ( i i ) 9 i s minimal - 9|a(x) (W) implies deg a(x) > deg 9 , ( i i i ) 9 i s monic. It i s shown ([2], theorem 4.2) that i f a key polynomial 9 over W i s assigned a value jx = W 9 > W9 , then the function W1 on R[xl defined by W\u00C2\u00BBf(x) = min {wfi(x) + i u j , 4. .where f(x) = f n ( x ) 9 + f n _ i ( x ) 9 ~ + + f 0 ( x ) , d e S fi(x) < deg 9 , i s a valuation of R [x]. Further, W 2 chosen above w i l l define a valuation V2 on R[x] i f assigned the value ^2 * V < p 2 > V l * 2 * T h e v a l u e v 2 satisfies V 2 V and and V 2f(x) \u00C2\u00AB V xf(x) = Vf(x) for a l l f(x) e R[x} such that deg f (x) < deg 92 \u00E2\u0080\u00A2 The second-stage value V2. i s .symbolized by v*2 = f V 0 , V^x \u00E2\u0080\u00A2 H I , V 292 \" ^2]* A s befbre, either V2 \u00E2\u0080\u00A2\u00C2\u00BB V or there exists a monic polynomial 93 of minimum degree satis-fying V2s>3 < V 9 3 . Again, i f 93 exists it i s a key poly-nomial over V2 and may be used to define a valuation V3 such that V3 < V and V 3f(x) = V 2f(x) = Vf(x) for a l l f(x) e R[xl with deg f (x) < deg 93 \u00E2\u0080\u00A2 The third-stage value V3 i s sym-bolized by V3 \u00E2\u0080\u00A2 [V 0, Yix = [ i i , V292 = |A2\u00C2\u00BB V393 \u00E2\u0080\u00A2 u.3]. MacLane shows ( [ 2 ] , theorem 8,1) that i f t h i s procedure i s continued, equality w i l l occur after either a f i n i t e or countable number of steps* In the f i r s t case V w i l l have a representation V - V k - [V 0, V 1 X = M-1,V292 8 8 ^ 2 \u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 1 vk \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 1 vk\u00C2\u00ABPk = Pk> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2]\u00C2\u00BB where Voof(x) .* lim V kf(x), and V is called a limit value* k-\u00C2\u00BB\u00C2\u00B0o Hence each valuation of Rfx] may be represented by one of these two cases i f every valuation of R is discrete* The key polynomials defining the above inductive and limit values satisfy: 5\" #iv) ~ V\u00C2\u00B1-l i s f a ^ 8 e \u00C2\u00A3 o r a 1 1 i > 2 ; (v) deg ^ > deg 9 i _ i f o r a 1 1 i - 2 \u00E2\u0080\u00A2 Now since every valuation of R T x ] is either an inductive value or the limit of a sequence of inductive values i t i s necessary, only to consider key polynomials which also satisfy (iv) and (v) \u00E2\u0080\u00A2 For this reason i t w i l l be assumed i n this paper that a key polynomial is a polynomial satisfying ( i ) , ( i i ) , ( i i i ) , (iv) and (v) . The representation of a valuation on R [x] is not necessarily unique, but i f one additional restriction is placed on the key polynomials the representation becomes unique when V0 on R is discrete. Let Vk= [V 0, V-jx =\u00E2\u0080\u00A2 u^, ... , V k 9 k ' \u00E2\u0080\u00A2 J AJ be an inductive valuation of R \u00C2\u00A3x]. The ? K value of f(x) e R [x] is found from the expansion f(x) - f n ( x ) 9 * + f n - l ( x ) ^k\" 1 + U - + f o ( x ) > where deg f^(x) < deg 9 k . By expanding each f^(x) i n powers of some integer N . Note: \"Pseudo-limit\" as defined here is not the same as that defined by Ostrowski. Now V 0 ( a i - a\u00C2\u00B1+i) = i * ^ , where )f\u00C2\u00B1 < ^\u00C2\u00B1+\u00C2\u00B1 for i > some integer N'. Since Y'I \u00E2\u0080\u00A2 v o ^ a i \" a i + l ) = vo[jai - a) + (a - a ^ i ) ] i t follows that, for i > N , - _ i . The pseudo-convergent sequences are now divided into two classes: , (1) fai} possesses a pseudo-limit i n A , (2) {OL\u00C2\u00B1\ does not possess a pseudo-limit i n A , Theorem 2.2: If the pseudo-convergent sequence {o.\u00C2\u00B1\ t with res-pect to V 0, has a pseudo-limit a E A , then the Ostrowski valuation V of A [x} defined by is the same- as the f i r s t stage valuation V, defined by' V, =fv6 , V, (x - a) * \u00C2\u00A5\ , where \u00E2\u0080\u00A2 lim \u00E2\u0080\u00A2 lim V0 (aj - aj + l ) . \u00C2\u00B1-tOO i - K \u00C2\u00BB Proof: It is sufficient to consider a monic linear polynomial x - /3 i n A[x}. Since - /3 \u00C2\u00BB (a^ - a) f: {a -/* ) and V G(ai - a) - T\u00C2\u00B1t either V Q (a \u00C2\u00B1 - ( 3 ) * J*i or, V ^ . - /3 ) = \ (a - ), for i sufficiently large. a. l i m I i i - . m a Hence V(x -f3) = lim V 0( a A - /3 ) \u00C2\u00AB min { T, V 0(a - /* ) } \u00C2\u00AB V^x -/?) i-*oo Theorem 2.3: Given a finite inductive value V \u00C2\u00AB [Yo,V(x - a) = **] on A [xj , a pseudo-convergent sequence \a^\ with pseudo-limit a e A can be found such that r \u00C2\u00AB lim Yi Proof: Let a / 0 i n A be chosen such that Va = d > 0 . Then there exists a real number o* such that cd = Jf* . A sequence of integers {n\u00C2\u00B1\ can be found such that ni no tit -\u00C2\u00B1 < < ... < - i < ... 10 10 2 10 1 and i-*oo 1 G i Let ft be any one of the roots of x - a . Then d = Va - V/^i =10 Yft or V ft = 1/10 d . Hence, the se-quence { v / ^ \" 4 } i s a s t r i c t l y increasing sequence with lim V ft*= lim \u00C2\u00A3 1 d = crd - Y . Let ax be defined by \u00E2\u0080\u00A2 ft + a \u00E2\u0080\u00A2 Since V(a; - a i + l ) * V(/9i W i- ft?) - 1ft1, {04} i s a pseudo-convergent sequence and since V( N . Therljfore, from V\u00C2\u00BB(x- ft) = limfmin/ft, VQ(*\u00C2\u00B1 - P)}] it. follows that lim Xi < V a i - P ) for i > N . Hence, V 0 ( a i + 1 - p ) = V 0 [ ( a i + 1 - a i ) + {a\u00C2\u00B1 - p )] \u00C2\u00AB X\u00C2\u00B1 < lim X i , for i > N , \"which is a contradic-i-\u00C2\u00BBoo tion. Thefefore V* - V . Suppose now that VT i s a Mac_ane valuation* From x - \u00C2\u00BB (x - a^) + (a i - i t follows that V-,^ - a i + 1,)' > V\u00C2\u00B1[x - a\u00C2\u00B1) \u00C2\u00BB X\u00C2\u00B1, for otherwise Vi[(x - a i + 1 ) - (a \u00C2\u00B1 HXjL+i) ] - V \u00C2\u00B1(x - a \u00C2\u00B1) - Y\u00C2\u00B1 > V 0 ( a \u00C2\u00B1 - a i + 1 ) and so x - a i + i ~ - a i + i ( v i ) \u00C2\u00BB which contradictsthe minimal .10 -condition ( i i ) of definition 1.4 for a key polynomial over V^. Therefore Y\u00C2\u00B1(x - a i + 1 ) = Y\u00C2\u00B1 and since l\u00C2\u00B1{x - a i + 1 ) < V i + 1 ( x - a i + 1 ) , &*i < *i+l f\u00C2\u00B0 r a l l i * 1 . Now V 0 ( a i - a i + 1 ) \u00C2\u00BB V'[(x - a i + 1 ) - (x - c^)] - Y i ; hence fai} i s a pseudo-convergent sequence. If ' ft were a limit of {a\u00C2\u00B1\, then lim VQia\u00C2\u00B1 - ft ) < \u00C2\u00BB . Let k > i be chosen such i\"*\u00C2\u00B0\u00C2\u00B0 that V$(a k : - ft ) > Y\u00C2\u00B1 , then V 0 ( a i - ft ) \u00C2\u00AB V 0 [ ( a i - a k) + (a k - ft )] = Therefore V\u00C2\u00BB(x - ft) = limlmin V (a\u00C2\u00B1 - ft )}1 = lim V 0(a. i -ft) = \u00C2\u00BB ; but V\u00C2\u00BB i s a f i n i t e value. Hence {a\u00C2\u00B1} has no limit in A .and w i l l , therefore, define an Ostrowski valuation which, by f i r s t part of theorem 2.4*. must be the same as V* . NOTE; If {a^} has a pseudo-limit a e A , then V\u00C2\u00BB may also be represented by L i-\u00C2\u00BBoo The results of this section now provide a connection be-tween the two methods of valuation A f x ] . In sections 4 and 5 i t w i l l be shown how a MacLane valua-tion of A[x] reduces to a MacLane valuation of R Tx] , that i s , the key polynomials and their assigned values w i l l be found for the reduced valuation on R fx}, and conversely how to extend a value on R fxj to A fx] \u00E2\u0080\u00A2 The connection between an Ostrowski valuation of R Lx] and a MacLane valuation of R Cxi w i l l then be clear. -11. 3 . The key polynomials defining the restriction of a valuation of A fx] to R [x] are intimately related to the key polynomials used by MacLane ( [3 ] ) to extend a valuation V 0 on R to a valuation W of R (a) , .a; separable extension of R \u00E2\u0080\u00A2 For this reason a description of the methods used by MacLane and the essential results w i l l now be given. As a particular example,consider the inductive value vk \" [ vo\u00C2\u00BB v l x Hl#2\u00C2\u00ABP2 e \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Vk'Pk18 He] of R TxJ and reassign to q>k the value + 0 0 . This defines a new, generalized valuation V = [ V V1 X = V2*2 \u00E2\u0080\u00A2 \u00C2\u00BB*2\u00C2\u00BB \u00C2\u00BB V l V l 8 8 hc-1* V < pk = \u00C2\u00B0\u00C2\u00B0] of R fx] \u00E2\u0080\u00A2 The generalized valuation V satisfies a l l the con-ditions of a valuation except that elements other than 0 are assigned the value +\u00C2\u00ABQ .. If a i s a root of (''k = \xA be an 12. inductive value of R [x] , If n n\u00E2\u0080\u00941 G(x) \u00C2\u00BB g n(x) 9k + g n - l ( x ) ?k~ + g\u00C2\u00A9 ( x) \u00C2\u00BB where deg gj^x) < deg 9 k , i s a polynomial i n R [ x i , then the projection of VK with respect to G(x) i s a -/2 , written proj (V k) \u00E2\u0080\u00A2 e - /3, where a and ft are the maximum and mini-mum values respectively of i such that VkG(x) \u00C2\u00AB V k[gi(x) 9k ] \u00E2\u0080\u00A2 Definition 3.2: The effective degree of G,(;x) in Uj>ko i s a : written D 0 . For i f proj (V^) \u00E2\u0080\u00A2 0 then would be only one term i n G(x) - anX11 .+ a ^ i x 1 1 \" 1 + ... + a 0 with minimum value, and VG(x) = V^Gix) \u00C2\u00A3 0 0 \u00E2\u0080\u00A2 To define a secon-stage value a monic polynomial f (x) of minimal degree satisfying Vf(x) > Vjf(x) is chosen. If f (x) i s not homogeneous in Vj, then i t s homogeneous part i s to be chosen. Denote this homogeneous part by 9 2 \u00E2\u0080\u00A2 As was mentioned i n the introduction, 9 2 i s a key 13. ^polynomial over \u00E2\u0080\u00A2 The second stage value V 2 is then defined by V 2 \u00E2\u0080\u00A2 \u00C2\u00A3 V q , V-^ = u^, ^2^2 8 8 ^ 2]* where p Vq>2 \u00E2\u0080\u00A2 Now* i f G(x) i s a homogeneous key over , then 93 i s chosen as G(x) and | i 2 b 0 \u00C2\u00B0 . That G(x) i s a monic polynomial of minimal degree satisfying VG(x) > V ^ x ) w i l l follow from lemmas 3\u00C2\u00AB3 and! 3.4. Lemma 3.3: Let V k be a k-th stage inductive value of R T x ] satisfying: (1) V k f ( x ) < Vf ( x ) for a l l f ( x ) e R [ x l , (2) deg f(x) < deg 9 k implies V k f ( x ) - Vf (x) , (3) \ minjvf, V[qfl}> min{v kf, \ l q $ \ because of (2), the choice of and the assumption (1) for q(x) . Hence ty|f(x) (V k) . ' Lemma 3.4: Let V k be an inductive value of R[x] \u00E2\u0080\u00A2 Any poly-nomial G(x) \u00C2\u00A3 R[x] has an equivalence decomposition G(x) ^ e(x) 9 k \u00C2\u00B0 t j 1 * 2 2 . . . flr (V k) , where each ty^ is a homogeneous key over V k , t Q > 0 and t\u00C2\u00A3 > G for 1 < i < r , and e(x) is an equivalence unit, that i s , D^etx) = 0 ifl V k . This decomposition i s unique ex-cept for equivalence units. Proof: Cf. [3J, theorem 4.2. 14. Now suppose q>2 * s a homogeneous monic polynomial of mini-ma}, degree satisfying Vi]. If G (x) i s not a homogeneous key over V^, then the second-stage i s given by v 2 8 8 [Vo\u00C2\u00BB v l x B 1^\u00C2\u00BB v 2?2 \" where V 2 < V . It is noticed again that proj (V 2) > 0 for otherwise VG(x) ^ \u00C2\u00B0\u00C2\u00B0 . Also 9 2 |G(X) ^ v l ^ ^ lemma MacLane's inductive process is repeated until G(x) does become a homo-geneous key over some V k or, i f this does not occur, i t i s re-peated a countable number of steps. In the former case v a [vo\u00C2\u00BB v l x e 1*1 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00E2\u0080\u00A2 V k \u00C2\u00B0 \u00C2\u00BB*k\u00C2\u00BB V G ( x > - \u00C2\u00B0\u00C2\u00B0] by the preceding argument. Also 9JL|G(X) ( V ^ _ I ) for a l l i such that 2 < i < k , and proj > 0 for 1 < i < k . If a countable number of steps are required, then V = V\u00C2\u00AB - [ v Q , V xx \u00C2\u00AB j i l f ... , V k 9 k = Hk, ...] \u00E2\u0080\u00A2 Certainly, \< V and 9 kJG(x) (v\" k - 1) for a l l k \u00C2\u00A3 2 . Since eaeh 9 k i s minimal over , deg 9 k < deg G(X) .. So from some point on a l l the keys w i l l have the same degree. In this case i t can be shown {[2], lemma 6.3) that the value group of Y f i s discrete i f the value group;; of & is discrete; that i s , the real numbers used as values for V form an isolated point set. If V f (x) < Vf (x) for some f (x) e E M , then V k f (x) < Vf (x) for a l l k > 0 by the monotone increasing character of the i n -1-5. ductive values and so

0 (Cf. the introduction). But sinee the value group of V1 i s discrete, Vf (x) > lim V k f (x) = oo . k-*\u00C2\u00BB Therefore only polynomials i n (G(x)) could satisfy V 0 for i > 0 and G for G(x) . If V 1_ 1 has been defined, the next key ^ i s chosen as any one of the tyj occurring i n the unique equivalence decomposition of lemma 3.4. The corresponding value i s chosen so that proj (V i) > 0 and u-i > Vj[_i\u00C2\u00ABpi \u00E2\u0080\u00A2 In the sequence of valuations so defined, each Vi i s called an i-th approximant to G(x) . MacLane not only shows that every such \"sequence of values defines a valuation *6. W on R(a) which i s an extension of V Q | but that : (1) i f G(x) eventually becomes a homogeneous key over unique for 2 < i < k and also, the value jx^ i s unique (f3] , theorem 5.3) . This implies that V 0 may be extended to R(a) i n only one way ([3l , theorem 10.1 ) , (2) i f a countable sequence of keys are required, then there i s at most a f i n i t e number of different sequences that can be constructed. Hence, V 0 on R may be ex-tended to R(a) i n at most a f i n i t e number of ways ([3l , theorem 10.1) . 4* The reduction, or re s t r i c t i o n , of an inductive value W of A[xl to Rtx] w i l l f i r s t be found; following theorem 4.7 the reduction of a limit value w i l l be found. These results w i l l be established by mathematical induction. Theorem 4.1: If W \u00E2\u0080\u00A2\u00E2\u0080\u00A2 [*W0, W(x - a) =)f] i s any inductive value of Afxl with W0a > Y , then W - W\u00C2\u00BB - [w0, Wfx - Y]. . Proof: Let x - ft e A[xl . I f W0(a - Z 3 ) < t, then W0(a - ft ) - We ft ; and the inductive value V k, then the i-th approximant i s *7. Theorem 4*3: Let W \u00E2\u0080\u00A2 [W q > W(x - a) - X\ be an inductive value of A f x ] with Y > WQa , then V X = [vo> V-jx = W0a] , where V Q \u00E2\u0080\u00A2 WQ on R , is the first-stage of the reduction of \u00C2\u00A5 to R[x] \u00E2\u0080\u00A2 There exist polynomials f(x) e R fx] such that V x f ( x ) < Wf(x) . Proof: The value of x i s Wx \u00E2\u0080\u00A2 min { Yt W 0 a } \u00C2\u00AB WQa . There-fore Wi = [wQ, Wjx * W0a] i s a first-stage value to \u00C2\u00A5 ; < W . Hence V - ^ - \u00C2\u00A3 V Q , V-jX - WQa] i s the first-stage of the reduction to Rfx] . Let G(x) * fx - a) (x - Z )^ ... (x - ftt) be the minimal polynomial of a i n Rfx} \u00E2\u0080\u00A2 Since W1(x - a) < W(x - a) and \ix < W(x - \u00C2\u00B1) , V 3 G U ) = W^ Gfec) < mix) .. Theorem 4.3 :shows that for Y > WQa at least one more key is necessary to obtain the correct reduction of W to Rfxl . Lemma 4.4: Let W = fwp, W(x - a) =Y] be any inductive value of A fx] \u00E2\u0080\u00A2 A polynomial f(x) \u00C2\u00A3 A f x l i s equivalence divisible by x - a i n W i f and only i f WQf(a) > Wf (x) . Proof: Let f(x) - f n ( x - a ) n + f n _ i ( x - a ) 1 1 \" 1 + ... + f 0 be the expansion of f (x) in powers of x - a ; f^ e A \u00E2\u0080\u00A2 Since Wf (x) \u00C2\u00AB min |wQf\u00C2\u00B1 + 1?}, W Qf 0 > Wf(x) ; and because f Q = f(a) , the :re_ation WGf(a) > Wf(x) always holds. Suppose W0f(a) > Wf(x). Then W [ f (x) - {f_(x - a ) n + ... + f x ( x - a)}] - W0f(a) > Wf(x) and, therefore, f(x) ~ f n ( x - a ) n + ... + f^x-a) in W ; that i s , x - a|f(x) i n W \u00E2\u0080\u00A2 Suppose, now, f (x) y q(x)(x - a) in W. Then fix) = q(x) (x - a) +.h(x) , where Wh(x) > Wf(x) \u00E2\u0080\u00A2 But, since h 0, the last term in the ex-pansion of h(x), i s f i t follows that 18. W 0f Q = W oh 0> Wh(x) > Wf(x) ; that i s , WQf(a) > Wf(x) . In the results to follow the polynomials... i and \i\u00C2\u00B1 defining the extension of this V 0 to the given W0 on R(a) . Theorem 4.5-s Let the polynomials > V k = ^ provided that x - aL. i n W \u00C2\u00B1 s false for a l l i i n the , \u00E2\u0080\u00A2 1 internal 1 <; i < k . Proof: By Theorem 4.3 and lemma 4 . 4 this theorem is true for k \u00C2\u00AB 1 \u00E2\u0080\u00A2 Suppose the result i s true up to k - 1 , then v k - l = [ Vo\u00C2\u00BB v l x 8 8 KL> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Vl'Pk-l = ^k-l] i s (k-Dst stage of the reduced value and V^ ., f (x) < Wf(x) for a l l f(x) e R f x ] . Let f(x) e R,fx_ be any polynomial such that deg f(x) < deg Wf(x) by lemma 4 . 4 , i t i s concluded that Wf(x) * V k_if(x) for a l l f(x) e R Tx] such that deg f (x) < deg i. Iswever, W9 k = W-^Oa) \u00E2\u0080\u00A2 p,k > V k _ i 9 i - t 19. since x - a|cpk in W i s false, and \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB 7 k - l V l - **k-l\u00C2\u00BB Vk B ^k] satisfies the relation V kf(x) < Wf(x) for a l l f (x) e R[xl and i s , therefore, a k-th stage of the restriction of W to RTx]. Lemma 4.6; If are the keys used to extend V 0 on R to W0 on R(a) , then there exists an i such that x - &) WQa , be .20. given on A[x] . Let {Vj.} be the sequence of approximants to G(x) , the minimal polynomial of a , defining the extension of V 0 on R to \u00C2\u00A5 0 on Ria) . If 9 k + i is the f i r s t key in these approximants such that x - a|?k+l(W) , then the reduction of W to R[x] i s given by: (1) V = [vo, V 1 X \u00C2\u00BB ui , ... , V k9 k = |xk, V9 k + 1 \u00C2\u00BB W9k+1] when V k 9 k + 1 < \u00C2\u00A5 9 k + 1' , (2) V k \u00C2\u00AB [v0, VjX \u00E2\u0080\u00A2\u00C2\u00BB p l f ... , V k9 k = u k] when V k + l 88 w\u00C2\u00A5k+l \u00E2\u0080\u00A2 Proof: As in theorem 4.5, V kf (x) - W0f(a) = Wf V k f D ( x ) and f Q(x)/vg(x) 9 k + 1(V K) which contradicts the minimal condition of the key .V\u00C2\u00BBf(x) ; for then .21 -V'f(x) = V\u00C2\u00BB[f(x) - f0U)] \u00C2\u00AB w[f(x) - f 0(x)] > min{wf(x), WfQ(x)} >V\u00C2\u00ABf(x) . Now w[f 0(x) - g(x) 9k+l] = Wflx) > V'f(x) \u00C2\u00BB V f f 0 ( x ) \u00C2\u00BB Wf0(x) ; hence f 0(x) ~ g(x)k+1(W), x - a|f0(x-)- (W) . But, since deg f Q ( x ) < deg w i ( x - ca) - *Y, . . . , w \u00C2\u00B1 ( x - ai) = ^ i ] can also be represented by \u00C2\u00A5^ = [w0, W ^ ( x - a^) \u00C2\u00BB ^ i ] . For, to find W ^ f (x) , f (x) i s expanded i n powers of x - and the coefficients are valued with W ^ ^ . But the coefficients are in A and are therefore actually valued by W Q . Lemma The value W \u00C2\u00B1 - \u00C2\u00A5 { + 1 - [ w 0 , W^ +j-tx -on A[x] . Proof: Let x - ft e A _xl . I f \u00C2\u00A5 0 (ai - ft ) < Y\u00C2\u00B1 then V a _ + i - ft) w o [ ( a i + i - a i ) + (*i -ft)] - w o ( a i - ft \u00C2\u00A5\u00C2\u00A3 + 1(x - P) = min/^ , W 0 ( a i + 1 m i n f ^ , - f t )} = \u00C2\u00A5i(x-/?) . If \u00C2\u00A5 0(ai - ft ) > ^ , then w 6 ^ a i + l - Z 3 ) > min f ^ , \u00C2\u00A5 0 ( a i - ft )} - ^ and w i + l ( x - ft ) W\u00C2\u00B1(x - ft ) . Lemma 4.9: For each x - ft e Atxl there exists a positive i n -teger N - such that \u00C2\u00A5(x - ft) = W^ (x - ft ) for a l l I > N . Proof: If no such N exists, then \u00C2\u00A5(x - ft ) > yj\u00C2\u00B1(x - ft ) for a l l i > 0 since \u00C2\u00A5 A < \u00C2\u00A5 i + 1 . Now \u00C2\u00A5 0 ( a i + 1 - ft ) > W ^ f x - ft) for otherwise from x - ft = (x - ai+i) + (ai+i - /* ) i t follows that y i i - \u00C2\u00A5 i(x - a i + 1 ) > w 0 ( a i + 1 - ft ) ; and therefore i+1 > w o ( a i + l - Z3 ) \u00E2\u0080\u00A2 Hence, \u00C2\u00A5(x - ft ) - \u00C2\u00A5 0 ( a i + 1 - /3 ) = \u00C2\u00A5 i + 1 ( x ) , which contradicts \u00C2\u00A5(x - ft ) > \u00C2\u00A5 i(x - ft ) for a l l i > 0 . Since \u00C2\u00A5 0(a i + 1-/S)>\u00C2\u00A5 i(x-/3)\u00C2\u00BB\u00C2\u00A5 i(x-a i+ 1) = Y\u00C2\u00B1i therefore 23. ^ W Q(a i + 1 - /3 ). W 0[(a i + 1 - a 1 + 2 ) + ( a i + 2 - / * ) ] = ^ i + i for a l l i > 0 . But this implies ft i s a psreudo-limit of M. Theorem 4.10: The k(i) keys occurring i n the reduction of Wj \u00E2\u0080\u00A2 [w#f'W|(x - ) on A[x] to R [ x l are the f i r s t k(i) keys i n the reduction of W = [w0, Wx(x - a x) - ^ , ... , Wi(x - c ^ ) \u00E2\u0080\u00A2 - \u00E2\u0080\u00A2 ] on ATxl to R [x3 Also, the values u w , for 1 < v < k ( i ) , i n the reduction of are the f i r s t k(i) - 1 values, for the keys i n the reduction of W \u00E2\u0080\u00A2 Proof: By lemma 4.9, -for any given f(x) e R [x] there exists an N . such that Wf (x) - Wjf (x) f o r a l l i > N . If V k ( i ) i s the reduction of Wj, then V k f i ) f(x) = Wf(x) for a l l i i i , Hence, the sequence of values fV^)} o n R txl gives every poly-nomial i n R\u00C2\u00A3x] the correct W value. It is. only necessary then to show that the k(i) keys in the reduction of are the f i r s t k(i) keys i n the reduction of W^ +, and that the values, with the possible exception of. Pk(i) \u00C2\u00BB are the same. By lemma 4.3, Wj_ and W i + 1=[W G, W i + 1(x- a i + 1 ) \u00C2\u00BB define the same valuation of A [ x j \u00E2\u0080\u00A2 Hence, they w i l l have the same reduction on R [x*] and, because the keys i n the reductions are homogeneous, each reduction w i l l be identical with respect to keys and values ([2], theorem 16.4) . As the value of x - <*i+i i s increased from to the valuet l* k(i) might increase and the keys, i f any, appearing i n Vjj.^^) but not i n V^d). are used to augment v k ( i ) t 0 vk(i+l) * These are the only changes that can happen; and at least one of these changes must happen. The truth of this follows from the discussion immediately after theorem 4.7 .24 and the fact that the minimal polynomial of 0^+1 definitely increases as the value of x - ai+i increases. Theorem 4.11: The reduction of a limit value W on Arx7 to R[x], as described i n theorem 4*10, is a limit value. Proof: Suppose the reduction is an inductive value V k * [ y o , V^x = Hi, ... , Vkq>k \u00E2\u0080\u00A2 |j,k] . By lemma 4 . 9 there exists a smallest i such that W^ q>k \u00C2\u00AB= Wk occur in the rep-resentation of V ; however, V may be an inductive value with keys past Wf(x) for a l l f(x) e A[x] . Further, the reduction of W\u00C2\u00B1 \u00E2\u0080\u00A2 [wOJ W^(x - a^) \u00E2\u0080\u00A2 ^i~\ i s Vi - f v 0 , Vix \u00E2\u0080\u00A2 (i i , ... , Vi min {w^ + iW'x}* Wxf(x) . The value Wx satisfies (2), (3), (4) and also (1) since 9 1 \u00E2\u0080\u00A2 x and ct^ = 0 . Assume theorem true up to W k-2.* Then Wk_i reduces to V k _ i and W f 9 k > V k _ ^ 9 k = W k - 1 9 k . For a key to augment Wk-1 l e t a factor x - a k of 9 k be chosen so that W1 (x-a k) > W k_i(x-a k) (Gf. introduction) and W(x - a k) \u00C2\u00A3 W\u00C2\u00BB (x - ft) for any factor x - ft of 9 k such that W' (x - ft ) > W^U - ft ) . Now define Wk by Wk \u00C2\u00BB [wk-1, Wk(x - a k) = Y k = W1 (x - a k ) ] . For x -ft , any factor of 9 k , Wk(x - ft) - min{Yk, W 0(a k - ft )) , and W'(x - ft) > min{y k, W 0(a k - ft )} . The inequality cannot hold; for, then, W(x - ft ) > Yk - Wk(x - a k) > Wk(x - ft ) > \ ^ ( x - ft ) , which contradicts the choice of x - a k . Therefore W\u00C2\u00BB (x - /3) m wk(x - ft) for a l l factors of ^ ; so l T 9 k = W k 9 k . Since 9k(<*k) \u00E2\u0080\u00A2 0 , certainly x - a k | 9 k i n Wk . 26. Jhis means the redaction w i l l use only keys y\u00C2\u00B1 for i < k . But Wk9k = Wfcpk = pk > Wk-l9k ,hence the reduction of must be . Theorem 5.2: If a value W\u00C2\u00BB = [ w o , W\u00C2\u00BB (x - ft) = s] on ACxl reduces to V k \u00E2\u0080\u00A2 \u00C2\u00A3 v o , V^x \u00E2\u0080\u00A2 M-I\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB vk^k ** ^ k\"] o n R C x l , then there exists an a , such that \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2) o n R t x l > then W* may be represented by W = \u00C2\u00A3Wq, Wjx = W2(x - a 2) \u00C2\u00AB Y 2 , ... , Wk(x - a k) \u00E2\u0080\u00A2 Yk .. where: (1) 9 k(a k) = 0 f or a l l k > 1 , (2) reduction of Wk to R fx] is Vfc , Proof: The proof i s similar to that for theorem 5.2. From theroems 5.2 and 5.3 i t i s seen that every valuation of A fx} may be put into a form such that each a k i s a root of the J27. -corresponding key cp^ appearing i n the reduction of the valua-tion to R [ x ] \u00E2\u0080\u00A2 This information indicates how a valuation V of Rtx] may be extended to some valuation of A fx] . It wi l l now be shown how this extension can be accomplished. Lemma 5.4: Let a be a root of some polynomial 9 e Atx] , then a valuation W \u00E2\u0080\u00A2 Jw o , W(x - a) \u00E2\u0080\u00A2 If] can be defined on A[x_ such that 9 has a prescribed value M> \u00E2\u0080\u00A2 The value J* i s uniquely determined* n n\u00E2\u0080\u00941 Proof: Let 9 = ft n ( x - a) + ft - a) + ... + ft^ix - a) , where ft\u00C2\u00B1 B k t then W9 = min | W0 ft \u00C2\u00B1+ i^}\u00C2\u00AB Let the numbers Y\u00C2\u00B1 be defined by WQfti + lY^ \u00E2\u0080\u00A2 jx for i = 1 , 2 , . . . . t m and defined by Y - max JTi . For this value of Jf, i WQ / ? i + i J* > jx for i \u00E2\u0080\u00A2 1 , 2 , ... , n and the equality holds for at least one value of i \u00E2\u0080\u00A2 Suppose there were two values Y and T , with Y > J*' > with the desired property. Since 9(a) = G , there exists an i ^ 0 such that I* - Wo ^ 1 + i y > WoA + 1 **' \u00E2\u0080\u00A2 So W defined by Y 1 would give 9 a value W9 < \i . There-fore the value W(x - a) i s unique. Lemma 5.5: In an inductive value Vk = [ Vo\u00C2\u00BB V l x 1 3 \"1> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB V l ^ k - l \" *k-tfk*k - \u00C2\u00BBk\ o f R W the for 1 < i < k are the complete, and only, set of ap-proximant s to 9 k \u00E2\u0080\u00A2 Proof: This follows immediately from [3] , theorem 5.3 (Cf \u00E2\u0080\u00A2 end of \u00C2\u00A73 ) i f V is defined by V 88 fVo\u00C2\u00BB V l x s \u00E2\u0080\u0094 \u00C2\u00BB v k - l * k - l = h c - l ^ k s .Theorem 5.6: Let WQ on A be an extension of V Q on R , then V k * [v0, V^x = Hi, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Vfc9k * o n R C x l may be extended to W * \u00C2\u00A3w0, W(x - a) on A[ X] , where 9k(<*) = 0* Proof: By lemma 5.4 the value W(x - a) = Y i s uniquely deter-mined from Wcpk = Hk \u00E2\u0080\u00A2 Now, W defined by this reduces to y/ = f vo\u00C2\u00BB v i x - ^i\u00C2\u00BB \u00C2\u00BB s *tr-i\u00C2\u00BB * v ] > where J < k , by theorem 4.7 since 9 k(a). ? 0 \u00E2\u0080\u00A2 But lAtyk = Hk and, so, V^ 1 9 k = M\u00E2\u0080\u009E 88 ^k^k \u00E2\u0080\u00A2 This implies yf \u00E2\u0080\u00A2 k and v \u00C2\u00BB Hk \u00E2\u0080\u00A2 That i s , the reduction of W is \ar W i s the extension of Vk \u00E2\u0080\u00A2 Since theorem 5.2 elaims every valuation W may be defined by some a where 9k (a) \u00E2\u0080\u00A2 0 , i t i s seen that for a given W0 the maximum number of extensions of V k on R Tx] to W on A Cxi i s the degree of \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB w k ^ x \" ak^ = *^k\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 on ACx] where: (1) ^(c^) = 0 for i = 1, 2, ... , (2) i s uniquely determined by the factor x - a i and the value Hi of 9i , (3) P^(x - a\u00C2\u00B1) =\u00C2\u00BB 1 in x - a u l for a l l i > 1 (Cf. definition 3.2) , (4) Wi reduces to Vi on R[x] for i - 1, 2, ... . Proof: It w i l l be possible to formally construct the sequence of values {Wi} i f i t can be. shown that there always exists a factor 29. of 9^ satisfying (3) . The construction of with this factor maybe accomplished by the method of theorem 5.6. How-ever, i t w i l l be necessary to show1 that W defined by this sequence of values is actually a MacLane value. In order to prove that property (3) may be satisfied i t w i l l be shown that, given a value WK which satisfies (1), (2), and (4) a value W k + i can be defined over WK also satisfying ( l ) , (2) and (4) and such that Dq,(x - a k + i ) =1 i n x - a k \u00E2\u0080\u00A2 Then, since satisfies (1), (2) and (4) i t follows by induction that property (3) can \u00E2\u0080\u00A2 be satisfied for a l l i > 1 . L e t WK satisfy (1), (2) and (4) and let W\u00C2\u00A3 be defined by WFC = [W Q , WJ(x - a k) - tf] with Y < yk but such that Vk-1k * y, < n k . Then as y \"* \u00C2\u00BB M- \"* l^ k \u00E2\u0080\u00A2 Therefore, i f 0 with respect to W k + i ( x - <*k+1) * ^k+l] where W K +^ satisfies O D , (2), (3) and (4) . It only remains to show that W defined by this sequence of values is a MacLane value. Since D 9(x - a k + 1 ) =1 i n x - a k , Y k - \u00C2\u00A5 k(x - a k) = W k(x - a k + 1 ) < W 0 ( a k - a k +^) ; that i s , J* k < W Q(a k \" ak+l^ \u00E2\u0080\u00A2 ^et wk+l D e defined by wk+l - [ wo\u00C2\u00BB w k f l ( x - a k + 1 ) = yk] , then WK = W K + 1 ; for, let x - ft e ACx] , then 30. W k( x - ft ) = min { Ykf W 0[a k - ft)] and w k + l > \" ft > - m i n { Y k , W 0(a k + 1 - ) } but V< ak +1 \" ft ) ^ min { w 0 ( a k + 1 - a k ) , W Q(a k - /3 )} > min { f k , W0(ak-/3)] Therefore W^ and w k + ^ have the same reduction on Rrx] \u00E2\u0080\u00A2 Hence, > i n order that W k +i reduce to V k +^ . Let Let W \u00C2\u00A9 \u00E2\u0080\u00A2 W W f +iW (Y . a ) -0*1 1 'k - V k k+l 9k ~ o 1! 1 k + l v x \u00C2\u00B0V * Therefore, since 1 ^ 0 , fk > Wk+1> w k + l ( x - ak>} \u00C2\u00BB b u t W k + 1(x - a k + 1 ) = y k+l > **k \u00C2\u00A3 w k + l ( x ~ ak>> a n d> s o \u00C2\u00BB w o K - ak+l> \u00E2\u0080\u00A2 w k + l ( x \" ak> ^ **k \u00E2\u0080\u00A2 But i t i s known from above that W0(ak - otk+i) \u00C2\u00A3 ^ . Therefore W Q(a k - a k + i ) \u00E2\u0080\u00A2 Y k and since $k+i > Yk for a l l k t 1 the sequence {ai} i s pseudo-convergent. The sequence has no pseudo-limit since W reduces to a limit value; otherwise V would be an inductive value ({2} , theorem 16.4) \u00E2\u0080\u00A2 There can be no limit for the sequence {aj.} i n A since V is a f i n i t e value; alter-natively, every limit of pseudo-convergent sequence i s a pseudo-li m i t . Therefore, W i s a MacLane valuation of A [x] satisfying properties (1), (2), (3) and Because of theorem 5\u00C2\u00BB3 and since for every limit value W of Afx], D "Thesis/Dissertation"@en . "10.14288/1.0080629"@en . "eng"@en . "Mathematics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Valuations of polynomial rings"@en . "Text"@en . "http://hdl.handle.net/2429/41331"@en .