( ), and the convex hull of A in ( K,K +

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1 Real cloed valuation ring Niel Schwartz Abtract The real cloed valuation ring, ie, convex ubring of real cloed field, form a proper ubcla of the cla of real cloed domain It i hown how one can recognize whether a real cloed domain i a valuation ring Thi lead to a characterization of the totally ordered domain whoe real cloure i a valuation ring Real cloure of totally ordered factor ring of coordinate ring of real algebraic varietie are very frequently valuation ring In particular, the real cloure of the coordinate ring of a curve i an SV-ring (ie, the factor ring modulo prime ideal are valuation ring) Real cloed valuation ring play a role in the definition of real cloed ring, a well a in the contruction of real cloure of ring and poring They can alo be ued for the tudy of univariate differentiable emi-algebraic function Thi lead to the notion of differentiablility of emi-algebraic function along half branche of curve Real cloed ring were introduced in [17] (cf [19]) and have been tudied ubequently in a large number of publication The cla of real cloed ring i cloed under the formation of reduced factor ring (cf [21], ection 12) Thu, factor ring of real cloed ring modulo prime ideal are real cloed domain Real cloed valuation ring, ie, convex ubring of real cloed field, were tudied in [4] and [5] in connection with ring of continuou function It i well-known that not every real cloed domain i a real cloed valuation ring (cf ection 1 for the contruction of example) The preent note tudie the quetion how one can recognize whether a real cloed domain i a real cloed valuation ring Theorem 11 give a characterization of thoe real cloed domain that are valuation ring Thi lead to a characterization of thoe totally ordered domain whoe real cloure i a valuation ring The criterion of Theorem 11 i not alway eay to check in a concrete etting, in particular in a geometric context Section 2 i devoted to totally ordered domain that arie in real algebraic geometry Partially ordered finitely generated algebra over totally ordered field are among the baic algebraic object in real geometry Totally ordered domain arie a reidue ring at point of the real pectrum So the following ituation will be tudied: There are a totally ordered field k,k + and a totally ordered integral k-algebra ( A, A + ) (which mean, in particular, that k +! A + " A + ); the trancendence degree trdeg k A i finite The quotient field of A i a totally ordered field, K,K +, and the convex hull of A in ( K,K + ) i a valuation ring Thi valuation ring yield numerical invariant that contain detailed information about the trancendence degree The numerical invariant lead to a charactization of thoe domain whoe real cloure i a valuation ring (Theorem 24) The numerical invariant are not alway eay to compute But they lead to practicable ufficient condition for the real cloure to be a valuation ring (Corollary 25) It follow that the real cloure i alway a valuation ring if the trancendence degree i at mot 1 (Corollary 26) In the cae of trancendence degree 2 there are real cloed domain that are not valuation ring, but they can be decribed completely (Example 28) Suppoe now that the field k i real cloed and that X i a real algebraic variety, the coordinate ring i denoted by k[ X] The real pectrum of k[ X] i partitioned into two ubet: P val ( X) (valuation point) i the et of prime cone! uch that the real cloure of the totally ordered domain k X [ ]! i a valuation ring; P val ( X) are the other prime cone If page 1 of 19

2 ! ", but P val ( X) i dene in the real pectrum in a very trong dim X > 1 then P val X ene, ie, with repect to a topology of the real pectrum that i much finer than the contructible topology (Propoition 210) The final ection i devoted to a dicuion of the role of real cloed valuation ring in the tudy of continuity and differentiability of emi-algebraic function Concerning continuity, thi i motly a brief reminder of earlier reult The valuation theoretic decription of continuity lead to a characterization of the continuou univariate emi-algebraic function that have one-ided derivative at a point (Theorem 31) The univariate reult lead to a theory of differentiability of emi-algebraic function and map along curve branche Thi i a very general notion of differentiability The cla of differentiable function on an affine pace over a real cloed field i the ame a the cla of continuou emi-algebraic function that atify a Lipchitz condition around every point (Theorem 36) To tart with, the notion of differentiability i developed for geometric ituation, ie, for emi-algebraic function defined on affine pace over real cloed field The method developed in [21] are ued to extend the notion of differentiability to abtract emi-algebraic function defined over arbitrary reduced poring (Theorem 39) Unexplained notation and terminology, in particular with regard to partially ordered ring and the real pectrum, can be found in [21] 1 Real cloure of totally ordered domain Convex ubring of totally ordered field are valuation ring, and convex ubring of real cloed ring are real cloed ring Real cloed field are alo real cloed a ring Hence convex ubring of real cloed field are real cloed valuation ring Thi i certainly an important and ueful cla of real cloed domain, but there are other real cloed domain a well Here i a method for contructing uch ring Suppoe that f : A! B i a homomorphim of real cloed ring and that C! B i a real! A " B C i a real cloed ring a well (ince the category of real cloed ubring Then f!1 C cloed ring i complete, cf [21], Definition 122, Propoition 27) The method yield real cloed domain that are not valuation ring: It ha been noted already that convex ubring of real cloed ring are real cloed Firt uppoe that S i a non-archimedean real cloed field, and let B! S be a convex ubring Let! : B " S B = B m B be the reidue map Note that S B i a real cloed field If L! S B i any real cloed ubfield then! "1 ( L) i a real cloed ubring of S It i a valuation ring if and only if L = S B Thu, chooing a proper real cloed ubfield of S B one obtain a real cloed domain that i not a valuation ring More generally, intead of a real cloed ubfield L one can ue any real cloed ubring C! S B to obtain a real # S The ring! "1 ( C) i a valuation ring if and only if C i a convex cloed ubring! "1 C ubring of S B If A i any real cloed domain with finite Krull dimenion d then A can be contructed recurively from real cloed field uing the method of the previou paragraph Thi i baed on the following obervation: The mallet non-trivial convex prime ideal p! A i a convex ubgroup of the quotient field S of A ([18], Lemma 8) Hence there i a larget proper convex ubring B! S, and p i the maximal ideal of B The reidue field B p i real cloed, and A p! B p i a real cloed ubring with Krull dimenion d! 1 The notation that i et up in the following paragraph will be ued throughout ection 1 and 2 page 2 of 19

3 Suppoe that A, A + i a totally ordered domain and ( K,K + ) i the totally ordered quotient field The upport map from the real pectrum to the prime pectrum i injective It image i the et of convex prime ideal, which i a totally ordered et and contain the ideal ( 0) There i a larget convex prime ideal, ay m A Uually the poring i not local But if it ha bounded inverion ([14], 7) then all maximal ideal are convex, hence the ring i local The convex hull of A in K i a convex valuation ring, denoted by It maximal ideal, m VK, i convex, and m VK! A = m A Thu, i alo the convex hull of A ma in K The extended ideal m A! i convex, but need not be a prime ideal However it radical i a convex prime ideal, and the localization W K = ma! i another convex ubring of K Thi i the larget convex ubring of K whoe maximal ideal retrict to m A In general the convex ubring and W K do not coincide Then R i the quotient field of!( A, A + ), the The convex hull of A in R i denoted by V R, the larget convex Now let R be the real cloure of K,K + real cloure of A, A + ubring of R that retrict to m A i denoted by W R Note that = V R! K and W K = W R! K The canonical map Sper! A, A + both homeomorphim The real cloure of A, A + ( ) " Sper( A, A + ) and the upport map of!( A, A + ) are + and of it localization ( A ma, A ma ) coincide So, if one tudie real cloure then one may uually aume that the totally ordered domain ha bounded inverion, ie, i local with convex maximal ideal m A The firt reult i a criterion to decide whether a real cloed domain i a valuation ring THEOREM 11 Suppoe that A i a real cloed domain with quotient field R The following condition are equivalent: (a) A i a valuation ring (b) A = V R (c) The incluion! : A " V R i an epimorphim in the category PO N PROOF (a) (b) The real cloed domain A i dominated by the valuation ring V R Valuation ring are maximal with repect to domination So, if A itelf i a valuation ring then A = V R (b) (a) and (b) (c) are trivial (c) (b) The map Sper! i injective by [21], Theorem 52 Since V R i the convex hull of A, it i clear that m VR! A = m A, and Sper! map the cloed point of Sper( V R ) to the cloed point of Sper( A) The functorial map between real pectra i alway convex ([12], Lemma 21), hence Sper! i bijective, even a implie that the functorial map :! ( A) #! ( V R ) between ring of emi-algebraic function ([21], Corollary 74) i an homeomorphim Surjectivity of Sper!! " iomorphim ([21], Corollary 715) Thu, V R i an intermediate poring between A and! A, and [21], Theorem 1213 how that V R! "( A) = A Ω There are everal immediate conequence of Theorem 11 The firt one i a characterization of thoe totally ordered domain whoe real cloure i a valuation ring: page 3 of 19

4 COROLLARY 12 Suppoe that ( A, A + ) i a totally ordered domain Then the following two condition are equivalent: (a) The real cloure!( A, A + ) of ( A, A + ) i a valuation ring (b) The canonical map! :( A, A + ) " V R i an epimorphim in the category PO N If thi i the cae, then!( A, A + ) = V R PROOF Note that the real cloure of ( A, A + + ) and ( A ma, A ma ), a well a the convex hull of both poring in the real cloed field R, coincide and that the canonical map ( A, A + + )! ( A ma, A ma ) i an epimorphim Therefore one may aume that A i local with maximal ideal m A (a) (b) The map! A,A + i alway an epimorphim in PO N (ince the real cloure i a monoreflector) The convex hull of A in R coincide with the convex hull of!( A, A + ) in R Theorem 11 how that!( A, A + ) = V R Thu the map! and! A,A + coincide, and the claim follow (b) (a) The convex valuation ring V R i real cloed The functorial propertie of the real cloure imply that! extend uniquely to a monomorphim! : " A, A + # V R, ie, one may conider!( A, A + ) a a ubring of V R Since! i an epimorphim and ince! =!! " A,A + implie that!( A, A + ) = V R Ω it follw that! i an epimorphim a well Theorem 11 COROLLARY 13 If A i a valuation ring (not necearily convex) then it real cloure! A, A + i a valuation ring PROOF If A i a valuation ring, then o i every localization of A Thu, A ma i a convex valuation ring of K,K +, and!( A, A + + ) =!( A ma, A ma ) So one may aume that A i a " V R i an epimorphim One convex valuation ring It uffice to prove that! : A, A + check eaily that the criterion of [21], Theorem 52, i atified Ω COROLLARY 14 If the real cloure of A, A + i an algebraic extenion of qf ( A m A ) i a valuation ring then V R = W R, and V R m VR PROOF Aume that V R! W R Then m WR! m VR! V R are ditinct convex prime ideal, and both retrict to the maximal convex prime ideal! Sper A, A + m A! A The functorial map i not injective, and ( A, A + )! V R i not an epimorphim in the Sper V R category PO N ([21], Theorem 52) Thi i a contradiction, and the firt claim ha been proved For the econd claim, let! be the cloed point of Sper A, A + cloed point of Sper! A, A + iomorphic Note that! " = V R Ω! A, A + and let! be the ( ) Then the real cloed reidue field!(" ) and!(") are i the real cloure of the totally ordered field qf ( A m A ) and that page 4 of 19

5 2 Totally ordered algebra of finite trancendence degree The preceding ection wa concerned with real cloure of arbitrary totally ordered ring Totally ordered ring that arie from coordinate ring of real algebraic varietie have pecial propertie Suppoe that S i a real cloed field and that! "Sper S X ( [ ]), where [ ] = S[ X 1,, X n ] The reidue ring S[ X]! i a totally ordered S-algebra with finite S X Krull dimenion and finite trancendence degree The real cloure of uch ring are frequently, but not alway, valuation ring It will be hown that the et of point! uch that! S X ( [ ] " ) i a valuation ring i very dene in Sper( S[ X] ) The method and reult are not limited to coordinate ring of varietie Rather, the deciive aumption i that one deal with algebra of finite trancendence degree over totally ordered field Since the focu i on real cloure it will be aumed that the totally odered ring have bounded inverion Thu totally ordered ring are local; the reidue cla ring modulo maximal ideal are totally ordered and, hence, have characteritic 0 Suppoe that B i a local ring with maximal ideal m B If char( B m B ) = 0 then B contain a ubfield k that i maximal with repect to incluion (Zorn' Lemma) One conider k a a ubfield of B m B via the reidue homomorphim! mb : B " B m B The extenion k! B m B i algebraic The field k i called a field of repreentative if k i mapped iomorphically onto B m B ([2], p AC IX 29) If thi i the cae then k i unique up to iomorphim Totally ordered ring with bounded inverion do not alway have a field of repreentative However: PROPOSITION 21 Every real cloed domain ha a field of repreentative PROOF Let k be a maximal ubfield of the real cloed domain A (note that A i local) It i claimed that k i a field of repreentative The total order of A retrict to a total order of k The univeral property of the real cloure reflector yield a k-homomorphim! k " A The, real cloure of a field i a real cloed field Therefore maximality of k implie that k =! k ie, k i real cloed A the extenion k! A m A i algebraic and A m A i a real cloed field a well one conclude that k i a field of repreentative Ω If B! C are domain then one define the trancendence degree of C over B by trdeg B C = trdeg qf( B) qf ( C) If C i a local domain again and k i a maximal ubfield then trdeg k ( C) i independent from the choice of the maximal ubfield Now conider the totally ordered domain A, A + again and uppoe that bounded inverion hold Then A i local and contain a maximal ubfield k Both k and A m A are totally ordered with poitive cone k + = A +! k and! ma A +, and ( k,k + ) i a totally ordered ubfield of A m A,! ma ( A + ) Thu every totally ordered domain with bounded inverion i an algebra over a totally ordered field The general notation and aumption of ection 1 are upplemented by the following hypothee: k, k + i a totally ordered field, and ha bounded inverion, and i a ( k,k + ) -algebra of finite trancendence degree A, A + A, A + It follow immediately from [15], Theorem 14C, that dim A + trdeg k A m A! trdeg k A A all chain of prime ideal of A are finite, thi i, in particular, true for the chain of convex page 5 of 19

6 prime ideal, ay 0 = p 0!! p = m A There may be prime ideal that are not convex, i equal to, hence o are the dimenion of ( ) and Spec (!( A, A + )) Let ( 0) = q 0!! q = m " A,A + be the chain of all Now valuation theoretic method will be ued to aociate numerical invariant with the and with it convex ideal The mallet convex prime ideal in that hence! dim A The dimenion of Sper A, A + Sper! A, A + prime ideal of! A, A + poring A, A + contain p 1 i denoted by p 1,VK ; the localization! = p1,v K i the larget convex ubring of K whoe maximal ideal retrict to p 1 in A Let! VK " : K # $ % VK " be the valuation aociated with! The value group i an abelian group of finite rank, and the valuation theoretic dimenion inequality yield rk! VK" = rk! " = r( A p i!1, p i p i!1 ) firt define r A, p 1 + trdeg k " m VK " # trdeg k A ([9], Theorem 343) One ( VK ) and then extend the definition to the other convex prime ideal: r A, p i Inide the valuation ring! there i a maximal ubfield k VK! that contain k Then trdeg k A = trdeg k V! = trdeg k k VK! + trdeg k! Note that trdeg k k VK! = trdeg k! m VK!, hence rk! VK" # trdeg k "! degree one obtain another erie of numerical invariant: t A, p 1 = t ( A p i!1, p i p i!1 ) " From the trancendence = trdeg k!! and t A, p i The following lemma record ome imple baic propertie of the numerical invariant defined above The eay proof are omitted: LEMMA 22 (a) trdeg k A p 1! trdeg k " m VK " (b) 1! r A, p i! t ( A, p i ) = r A p j, p i p j (c) If j! i " 1 then r A, p i One conclude that and induction yield and t A, p i Ω = t A p j, p i p j t ( A, p 1 ) + trdeg k A p 1! t ( A, p 1 ) + trdeg k " m VK " = trdeg k A,! t A, p i + trdeg k A p " t A, p 1 i=1 The following inequality follow trivially from the Lemma: + trdeg k A p 1 " trdeg k A + trdeg k A p! " r A, p i + trdeg k A p! " t A, p i + trdeg k A p i=1 LEMMA 23 (a) trdeg k A p i = trdeg k! A (b) r( A, p i ) = r (!( A),q i ) q i i=1 page 6 of 19

7 = t (!( A),q i ) Ω (c) t A, p i The numerical invariant lead to another characterization of totally ordered domain whoe real cloure i a valuation ring: THEOREM 24 The following tatement about the totally ordered k, k + equivalent: (a)! A, A + i a valuation ring # & (b) trdeg k A! " t ( A, p i ) + trdeg k ( A p ) $ % ' ( = 0 i=1 -algebra ( A, A + ) are PROOF All numerical invariant are preerved by tranition to the real cloure So one may aume that A i real cloed, and then tatement (a) ay that A i a valuation ring (a) (b) If A i a valuation ring then A = V R, and the factor ring A p i are convex ubring of real cloed field a well Condition (b) follow (by induction) from the equality trdeg k A = t ( A, p 1 ) + trdeg k A p 1 (b) (a) The proof i by induction on If = 0 then A i a real cloed field, and the claim i true Suppoe that the implication i true for! 1 The mallet prime ideal p 1 i convex in R, hence it i alo the mallet nontrivial prime ideal of the valuation ring V R Condition (b) implie Since t V R, p 1 trdeg k A =! t A, p i + trdeg k A p i=1 + trdeg k A p 1 " t ( V R, p 1 ) + trdeg k V R p 1 = trdeg k A " t A, p 1 = t ( A, p 1 ) (by definition) one conclude that trdeg k A p 1 = trdeg k V R p 1 = qf ( V R p 1 ), and V R p 1 i the convex hull of A p 1 in qf ( A p 1 ) Then qf A p 1 Furthermore, dim A p 1 =! 1 and $ ' 0! trdeg k A p 1 " # t ( A p 1, p i p 1 ) + trdeg k A p % & ( ) i=2 " t A, p 1 = trdeg k A p 1 + t ( A, p 1 ) $ % & ' # + trdeg k A p ( ) + t ( A p 1, p i p 1 ) i=2 $ '! trdeg k A " # t ( A, p i ) + trdeg k A p % & ( ) = 0 i=1 Induction how that A p 1 = V R p 1, and A = V R follow immediately Ω There are everal immediate conequence of the Theorem: COROLLARY 25 The following condition both imply that! A, A + = 0 (a) trdeg k A! + trdeg k A p # & (b) trdeg k A! " r( A, p i ) + trdeg k A p $ % ' ( = 0 i=1 i a valuation ring: page 7 of 19

8 PROOF The inequality " trdeg k A! r( A, p i ) trdeg k A! + trdeg k A p and Theorem 24 imply the claim Ω $ ' # + trdeg k A p % & ( ) i=1 $ ' " trdeg k A! # t ( A, p i ) + trdeg k A p % & ( ) " 0 COROLLARY 26 Suppoe that the domain A and the field k are real cloed If trdeg k A! 1 then A i a valuation ring PROOF If dim A = 0 then A i a real cloed field, and there i nothing to prove If dim A = 1 then Corollary 25 (a) applie Ω The condition of Corollary 25 are not neceary condition for!( A, A + ) to be a valuation ring, a the following example how EXAMPLE 27 Let K =! X the total order on K that i compatible with v and atifie 0 < X (cf [13], p 72) The real cloure of K,! i=1 and define a real valuation v : K! "! by v( X) = 1 Let! be i denoted by R Let v R be the natural valuation of R The value group i () ([16], p 62, () of valued field i immediate and proper and ha () \ R and conider the algebraic in!((")) The natural valuation ring of R and S are denoted by A and B! There i an embedding of R into the generalized power erie field! " Satz 21) The extenion R!! " infinite trancendence degree Pick any element t!! " cloure S of R t Both are real cloed ring of dimenion 1 The extenion A! B i immediate a well All data concerning only the reidue field or only the value group are the ame, hence On the other hand, dim A + trdeg! A m A = dim B + trdeg! B m B = 1, + trdeg! A m A = r( B,m B ) + trdeg! B m B = 1 r A, m A = 0, trdeg! B! ( dim B + trdeg! B m B ) = 1, + trdeg! A m A + trdeg! B m B trdeg! A! dim A + trdeg! A m A trdeg! A! r A, m A = 0, trdeg! B! ( r B,m B ) = 1 Ω The tatement of Corollary 26 doe not extend to real cloed domain with trancendence degree larger than 1 The following example give a complete decription of real cloed domain that have trancendence degree 2 over a real cloed field: EXAMPLE 28 Let A be a real cloed domain with quotient field R that ha trancendence degree 2 over a real cloed field k Let l be a field of repreentative that contain k If k! l then trdeg l ( A)! 1, and A i a valuation ring (Corollary 26) So uppoe now that k i a field of repreentative and that A i not a valuation ring Let p 1! A be the mallet nontrivial prime ideal of A Then p 1 i alo the mallet nontrivial prime ideal of V R If A p 1 ha trancendence degree 1 over k then it i a valuation ring There are two poibilitie: page 8 of 19

9 If A p 1 i a real cloed field then k i not a field of repreentative, a contradiction If A p 1 i a proper convex valuation ring in it quotient field then it coincide with V R p 1, and one conclude that A = V R, which contradict the aumption that A i not a valuation ring Thu it ha been hown that p 1 mut be the maximal ideal of A There are two cae again: (i) The maximal ideal p 1 of A i alo maximal in V R, and V R p 1 i a real cloed extenion of k with trancendence degree 1; the extenion i Archimedean over k (ii) The quotient field of V R p 1 i a real cloed extenion of k with trancendence degree 1, and V R p 1 i the convex hull of k in qf V R p 1 Both cae occur, a the following contruction how (although the firt cae i obviouly impoible if k =! ): (i) Let R be the real cloure of the ubfield R 0 (!,t) of the power erie field!((")) (where R 0 i the field of real algebraic number and t i an element with value 1) Let V R be the natural valuation ring of R, and define A = R 0 + m VR of the power erie field!(("! ")) = ( 1,0 ), = ( 0,1) Let V R be the natural valuation ring of R There are two non-trivial prime! p! m VR One define A = R 0 + p Ω (ii) Let R be the real cloure of the ubfield R 0,t with lexicographically ordered value group!!!, natural valuation v, and v v t ideal in V R : 0 The next example decribe, in the cae of trancendence degree 2, how the connection between a real cloed domain and it convex hull in the quotient field can be undertood geometrically uing a blowing-up procedure EXAMPLE 29 The ring C = CSA! 2 of continuou emi-algebraic function on the real [ ], partially ordered by the um of plane i the real cloure of the polynomial ring! X,Y quare It real pectrum i homeomorphic to the prime pectrum and to the real pectrum of the polynomial ring Conider the quadratic tranformation! :! X,Y [ ] "![ X, Z ], where Geometrically thi repreent a blowing-up The real cloure of![ X,Z ] i denoted Z = Y X by D The ring C and D are iomorphic, but the quadratic tranformation yield a non-trivial homomorphim! " following condition: 0 < X, Z, 0 :C # D Let! "Sper(![ X, Z ]) be the prime cone defined by the! ( X)! ( X,Z ) i the equence of convex prime ideal The correponding prime cone of D i denoted by! The real cloed domain D! i the real cloure of![ X, Z ]! The trancendence degree over! i 2, and there are two nontrivial convex prime ideal, hence Corollary 25 how that D! i a valuation ring Now conider! = " #1 ( $ ) and the correponding prime cone! "Sper( C) ; note that! = "(# ) $1 (% ) Then C! i the real cloure of![ X,Y ]! From Y = X! Z in![ X,Z ], it follow that ( X,Y ) i the only nontrivial! -convex prime ideal of![ X,Y ] Therefore C! ha Krull dimenion 1 The quotient field of C! and of D! coincide, and D! i the convex hull of C! in the quotient field, a well a the real cloure of C! [ Z ] Ω page 9 of 19

10 The conideration in Example 28 and Example 29 can be extended to algebra of higher trancendence degree in an obviou way Suppoe that the field k i real cloed, and conider a real algebraic variety X with coordinate ring k[ X] The prime cone! "Sper( k[ X] ) uch that!( k[ X] " ) i a valuation ring form a very large ubet in the real pectrum It contain all prime cone of dimenion 0 and 1 Moreover, if! i any prime cone and if there are finitely many function a 1,,a r!k X [ ] with a i (!) > 0 then there i a prime cone! with upp (! ) = upp (") and > 0 uch that!( k[ X] " ) a valuation ring: If!( k[ X] ") i a valuation ring then there a i! i nothing to prove So, aume that thi i not the cae Replacing X by the variety defined by upp (!) one may aume that upp (!) = ( 0) A X i irreducible, the open emi-algebraic { > 0} i Zariki-dene in X According to [6], Propoition et U = x!x "i = 1,,r : a i x 811, there are a point x!u and a pecialization chain with greatet poible length dim X in!u =! "Sper k X { ( [ ]) #i = 1,,r : a i (! ) > 0}, ay! 0 " "! dim X = x The point! 0 ( [ ] " 0 ) i a valuation ring one ue Corollary 25 k[ X] atifie the requirement (To ee that! k X (a) and note that dim X = trdeg k, cf [6], Propoition 82 and 83) Intuitively the dicuion how that the et of valuation point i very dene in the real pectrum Thi can be made precie if a uitable topology i introduced on the real pectrum: Let T be the topology of Sper k X with the et upp!1 ( [ ]) that i generated by the Harrion topology together ( p), p!spec( k[ X] ) Thi topology i finer than the contructible topology Hence, for it definition one can alo refer to the contructible topology intead of the Harrion topology Oberve that the definition of the topology T doe not ue in any way that one conider the coordinate ring of a variety; the ame definition can be made for the real pectrum of any ring Uing the topology T, the dicuion about denene of the et of valuation point amount to the following tatement: PROPOSITION 210 The et of valuation point in Sper( k[ X] ) i dene with repect to T Ω 3 Continuity and differentiability of emi-algebraic function The preent ection recall how valuation have been ued to define real cloed ring, ie, ring of continuou emi-algebraic function over poring Valuation can be ued to give a very imple characterization of differentiability of univariate emi-algebraic function Thi lead to a new notion of differentiability for continuou emi-algebraic function It i equivalent to the exitence of local Lipchitz condition The contruction i a priori geometric, but can be extended to an H-cloed reflector on the category of reduced poring (cf [21], ection 10) Real cloed pace were introduced in [17] a a vat generalization of emi-algebraic pace (cf [7]; [8]) Affine real cloed pace are pro-contructible ubet of real pectra together with a tructure heaf of ring The main tep in the contruction of real cloed pace i the decription of the ring of ection They are ring of contructible and compatible ection over pro-contructible et ([19], ection I2) Compatibility i a valuation-theoretic condition: Suppoe that K! Sper( A) i a pro-contructible ubet and that there i a emi-algebraic function a! $ "(# ) (ie, a contructible ection) If!," #K #!K with! "{#} then upp (!) " upp (# ), and the canonical map! ",# : A # $ A " i a page 10 of 19

11 homomorphim of totally ordered ring Suppoe that!(") i the real cloure of the totally ordered quotient field of A! There i alway at leat one convex ubring C uch that the maximal ideal m C retrict to upp (! ) upp (") in A! Among thee ubring there are a larget one, denoted by C!", with maximal ideal m!", and a mallet one, denoted by D!", of the localization of with maximal ideal n!" The mallet one i the convex hull in! " A! at the prime ideal upp (! ) upp (") The larget one i the union of all thee valuation ring The containment D!" # C!" i proper in general If C i one of thee valuation ring then there i a canonical map! C : A " # C m C, which extend uniquely to a homomorphim! C : " # $ C m C The canonical place of C i denoted by! C The "C #! and! C"# ( a (#)) = $ C"# ( a (" )) [ ], k i a real cloed field, X! k n i a real i a contructible ubet Then the ring of function a i compatible if it i alway true that a! Conider a geometric ituation, ie, if A = k X algebraic variety over k, and K! Sper A contructible and compatible ection over K i canonically iomorphic to the ring of continuou emi-algebraic function S! R, where S! X i the emi-algebraic ubet that correpond to K ([6]) The contructible ection correpond to the emi-algebraic function Compatibility expree continuity Suppoe that f : S! R i any emi-algebraic function Then the point x where f i dicontinuou form a emi-algebraic ubet T! S with dim x T < dim x S for all x!t Every non-iolated point x!s ha a generalization that belong to P val ( X) (by the Curve Selection Lemma, [7], or [1], Théorème 255) Therefore one can check continuity of f by conidering the compatibility condition only for point! "P val ( X) # K and pecialization! "S Then C!" i the real cloure of A!, and compatibility of the ection a! $ "(# ) can be rephraed in the following form now: a i bounded at! (ie, a! a converge to a! "C #,! ), and along! (ie,! "# ( a (#)) = a (" ) ) There i an intereting connection with a contruction propoed by Brumfiel: In [3] he intended to define real cloure of ring (and to ue them for the tudy of the reduced Witt ring aociated with ring) With any ring A he aociated the ring of contructible and continuou ection of the canonical map Sper A T #!K ( [ ])! Sper( A) of real pectra In general the ring contructed in thi way i properly larger than the real cloure of the ring A and lack the functorial propertie that are needed for the application A explained above, the real cloure of A i contructed a a ubring of $!(" ) Brumfiel' ring can be preented a a " #Sper A ubring of the ame product The condition of compatibility i replaced by a weaker valuation theoretic condition: A emi-algebraic function a! $ "(# ) belong to Brumfiel' ring if and only if a! #!Sper A = $ D"# ( a " ) "D #! and! D"# a (#) [ ] i the coordinate ring of a real variety Then Brumfiel' Now uppoe that A = k X contruction yield exactly the real cloure of A: The real cloure i the ring of continuou emi-algebraic function on X Aume by way of contradiction that Brumfiel' ring i properly larger than the real cloure Then it contain ome dicontinuou emi-algebraic function f If x i a point of dicontinuity then curve election yield a continuou emialgebraic curve branch! : 0," [ ) # X uch that! ( 0) = x and f!! i dicontinuou at 0 page 11 of 19

12 of the rational point The curve branch define a 1-dimenional generalization! "Sper A x!sper( A), hence a valuation point of the real pectrum The convex valuation ring D!" # $ " upp x i the only convex valuation ring whoe maximal ideal retrict to upp (!) in A! Therefore it i the ame ring a C!", and Brumfiel' condition coincide with compatibility, hence the function f!! cannot be dicontinuou at 0 Thi contradiction prove that Brumfiel' ring i indeed the real cloure in geometric ituation The Curve Selection Lemma implie that continuity of emi-algebraic function can be checked by conidering retriction to germ of curve branche A emi-algebraic function f : X! k defined on a emi-algebraic et i continuou if and only if, for all point x!x and for all continuou emi-algebraic map w : 0,! [ ) " [ 0,# ) uch that map f! w 0,! [ ) " X, with 0 <! "k and w( 0) = x, there i a ubinterval 0,! [ ) i continuou The length of the interval i immaterial, only continuity at 0 matter Note that any emi-algebraic map g :( 0,! ) " k i continuou on ome initial piece ( 0,! ) of the interval ( 0,! ) Therefore the following dicuion will focu on continuity of emi-algebraic function f : 0,! [ ) " k at 0 Let A = k[ T ] be the univariate polynomial ring over k Recall that proper pecialization in Sper( A) have the following form (cf [1], Example 714): The pecial point i an element z!k, and the general point i one of { : f ( t) > 0} : f ( t) > 0 z + = f!a "# > 0 $t! z,z + # { } z! = f "A #$ > 0 %t " z,z! $ The real cloure of A z + and A z! are the convex valuation ring C z,z+ and C z,z! Both valuation ring have the reidue field k The value group are iomorphic to!, though not canonically If one i intereted in local quetion then the only important fact about z i that it i a rational point Therefore one may aume frequently that z = 0 The valuation aociated with C 0,0+ i denoted by v = v 0,0+ : qf C 0,0+! " 0,0+ # $ i the correponding place Continuity of f [ 0,! ) at 0 mean that f 0 + ( ) = f 0 { } ;! =! 0,0+ :C 0,0+ " k!c 0,0+ and that! f 0 + It ha been hown in [21], Example 1411 that one can aociate a ring of continuouly differentiable emi-algebraic function with each partially ordered ring, exactly in the ame way a continuou emi-algebraic function lead to real cloed ring The quetion whether a given emi-algebraic function i continuou can be anwered by checking compatibility How can one recognize whether a emi-algebraic function i differentiable? There are variou different notion of differentiability for emi-algebraic function (cf [10]) They coincide for function of one variable Note that differentiability implie continuou differentiability, [10], Propoition 57 The firt reult i a characterization of differentiable emi-algebraic function of one variable The identity function of k i denoted by T A continuou emi-algebraic function f i differentiable at a point z!k if the continuou emi-algebraic function q f,z : x! f ( t) " f ( z) T ( t) " T ( z) the cae if and only if q f,z ( z! ) "C z,z!, q f,z z + defined on k \ { z} can be extended continuouly to the point z Thi i!c z,z+ and! z,z+ q f,z ( z " ) =! z,z+ ( q f,z ( z + )) The extended function q f,z belong to the real cloure!( A) of A, and page 12 of 19

13 f! f z " q f,z belong to the principle ideal of! A = T! T ( z) The value of q f,z at z i the derivative of f at z q f,z generated by T! T ( z) The function f i differentiable to the right (or to the left) of z if the function q f,z (!",z) ) can be extended continuouly to the cloed interval z,+! [ ) (or!",z ( z,+! ) (or ( ]) Thi i!c z,z+ (or q f,z ( z! ) "C z,z! ) The right (or left) derivative i the the cae if and only if q f,z z + value of the extended function at z The function f i differentiable at z if and only if it i differentiable on both ide and the one-ided derivative coincide Thee remark reduce quetion about differentiable function of one variable to quetion about one-ided differentiability at a ingle point Therefore it i important to be able to recognize one-ided differentiability at a point THEOREM 31 The emi-algebraic function f : 0,! and only if v( f ( 0 + )! f ( 0) ) " v( T ( 0 + )) [ ) " k i differentiable to the right of 0 if!c 0,0+,! f ( 0) = T ( 0 + ) " q f,z ( 0 + ) yield the deired inequality between the value PROOF Suppoe that f i differentiable at 0 from the right-hand ide Then q f,z 0 + and f 0 + Converely, uppoe that the inequality hold The coefficient field k i a field of repreentative for the valuation ring C 0,0+ Therefore there exit element a!k and b!m 0,0+ uch that ie, q f,z Ω = f ( 0 + )! f ( 0) T 0 + f ( 0 + )! f ( 0) = ( a + b) "T ( 0 + ), = a + b "C 0,0+ The element a i the right-ided derivative of f at = 1 For { " r}, J r = { a!c 0,0+ v( a) > r} The value group of v = v 0,0+ will be identified with! now in uch a way that v T r!! one define the valuation ideal I r = a!c 0,0+ v a For each valuation ideal, ay I, the ubring k + I! C 0,0+ i a local ring with maximal ideal I (cf the " D + M -contruction" in [11], Appendix 2) The emi-algebraic function f i differentiable to the right of 0 if and only if f ( 0 + )!k + I 1 The local decription of one-ided differentiability of univariate continuou emi-algebraic function can be extended to a theory of one-ided differentiability for general emi-algebraic function and map Informally peaking, one tart with one-ided differentiability at a point and add the condition that differentiable map are cloed under compoition A little more preciely, one proceed a follow: A curve branch w :[ 0,! ) " k n i aid to be differentiable at 0 if each component w i :[ 0,! ) " k i differentiable on the right-hand ide of 0 A continuou emi-algebraic function f : k n! k i differentiable at z!k n differentiable to the right of 0 for each differentiable curve branch w : 0,! if f! w i [ ) " k n with = z It i obviou that the emi-algebraic function that are differentiable at z form a w 0 ring (A decription of thi ring will be given below, Corollary 38) Finally, a emi-algebraic page 13 of 19

14 map! : k p " k n i differentiable at z!k p if the compoition f!! i differentiable at z for Up to thi point all definition are local Global notion are obtained in the obviou way: each emi-algebraic function f : k n! k that i differentiable at! z One demand that function and map are differentiable at every point Thi notion of differentiability ha not been dicued in the literature o far The poibility to define differentiability along continuouly differentiable curve ha been mentioned briefly in [10], Example 510 But the preent concept i different ince it peak only about one-ided differentiability It will be hown now that the new notion of local differentiability i equivalent to the exitence of a local Lipchitz condition For preparation, a couple of lemma are needed in order to better undertand the notion of differentiability outlined above LEMMA 32 The curve branch w : 0,! ome!, 0 <! " #, and ome 0 <! uch that w t [ ) " k n i differentiable at 0 if and only if there are! w( 0) " # $ t for all t![ 0," ) PROOF It i poible, of coure, that the curve branch i contant on an initial ub-interval [ 0,! ) " [ 0,# ) Then w i differentiable and w( t) = 0! t for all t![ 0," ) Suppoe now w that w i not contant near 0 If w i differentiable then the limit lim i ( t) " w i ( 0) t!0 = # i t w( t) " w( 0) n 2 n 2 exit, and lim t!0 = $ # i=1 i ; define! = 1 + "! i=1 i Continuity implie t that there i an interval [ 0,! ) on which! w( 0) t Converely, aume that! and! exit a tated For each component of w one obtain the etimate w i ( t)! w i 0 w lim i ( t) " w i 0 t!0 t w t " w( t)! w( 0) " # $t Thu w i ( t)! w i ( 0) t " #, ie, w( t)! w( 0) " # $ t i bounded near 0, and exit, which mean that the curve branch i differentiable at 0 Ω Local differentiability i defined uing compoition with curve branche Therefore it i important to undertand when two curve branche yield the ame information Claically thi i the cae if two curve branche are reparametrization of each other Here the ituation i eentially the ame Let! :[ 0," ) # [ 0,$ ) and! :[ 0," ) # [ 0,$ ) be two mutually invere continuou emialgebraic curve branche They define mutually invere iomorphim! * :C 0," ([ )) # C( [ 0,$ )) and! * :C([ 0," )) # C( [ 0,$ )) between the ring of continuou emi-algebraic function Localizing at 0 one obtain two mutually invere automorphim! 0 * :C 0,0+ " C 0,0+ and! 0 * :C 0,0+ " C 0,0+ The curve branche are local diffeomorphim at 0 if and only if both are differentiable at 0 (Recall, eg, from [7], that a continuou emialgebraic curve branch! : 0," [ ) # k with! ( 0) = 0 i either contant near 0 or i injective [ ) " [ 0,# ) In the econd cae! ([ 0," )) = [ 0,# ) i an interval, and [ ) $ [ 0,% ) i a continuou emi-algebraic map) on a mall interval 0,!! "1 : 0,# page 14 of 19

15 LEMMA 33 The curve branch! : 0," ( ( )) = 1 v! 0 * T 0 + [ ) # [ 0,$ ) i a local diffeomorphim at 0 if and only if ( ( )) = 1 Theorem 31 implie that! i differentiable at PROOF Firt uppoe that v! * 0 T Moreover,! i not contant near 0, and one may aume that! i bijective It follow that! 0 * and! "1 that v! "1 * 0 are mutually invere automorphim of C0,0+ Thu v (! * 0 ( T ( 0 + ))) = 1 implie * 0 ( T ( 0+ )) = 1 a well But then! "1 i differentiable at 0 a well, ie,! i a local diffeomorphim at 0 Converely, aume that! i a local diffeomorphm Then v! 0 * T 0 + v! "1 ( ( )) " 1 and * ( 0 ( T ( 0+ ))) # 1 A! * 0 and (! "1 * ) 0 are mutually invere automorphim of C0,0+ one conclude that both value are 1 Ω LEMMA 34 Suppoe that w 1 :[ 0,! 1 ) " [ 0,! 2 ) i a curve branch Then the following condition are equivalent (i) For all curve branche w 2 :[ 0,! 2 ) " k n (ome n) and all continuou emi-algebraic function f : k n! k, differentiability of f! w 2 i equivalent to differentiability of f! w 2! w 1 (ii) w 1 i a local diffeomorphim * 0 :C 0,0+! C 0,0+ * 0 ( 0 + )!k + I 1 if and only if * 0!k + I 1 (i) (ii) It i clear that w 1 cannot be PROOF (ii) (i) If w 1 i a local diffeomorphim then the automorphim w 1 retrict to an automorphim of k + I 1 Thu f! w 2 * * ( f! w 2! w 1 ) 0 ( 0 + ) = ( w 1 ) 0! f! w contant near 0 So one may aume that it i bijective, hence invertible Let w 2 = w!1 1 be the invere curve branch of w 1, and conider the emi-algebraic function f = w 1 A id = f! w 2 i differentiable it follow that w 1 = f! w 2! w 1 = w 1! w 2! w 1 i differentiable Next, let f = id Then id = f! w 2! w 1 i differentiable, hence w 1!1 = f! w 2 i differentiable Ω According to Lemma 34 two curve branche yield the ame information about differentiability if and only if they are related by a local diffeomorphim It i clear that, if a curve branch i a compoition w!! and if! i differentiable, differentiability of f! w implie differentiability of f! w!! The next lemma how that it uffice to conider curve branche w : 0,! [ ) " k n with w( t)! w( 0) = t in order to check differentiability of emialgebraic function: LEMMA 35 Let w : 0,! [ ) " k n be an injective curve branch Then there are an element!, [ ) " k n uch that w 1 ([ 0,! )) = w( [ 0," )) and ( )! w 1 ( 0) = for all![ 0," ) 0 <! " #, and a curve branch w 1 : 0,! w 1 PROOF The emi-algebraic map! : 0," [ ) # k :t # w( t) $ w( 0) i continuou,! ( 0) = 0, [ ) " [ 0,# ) on which! i and! i not contant near 0 Therefore there i a ubinterval 0,! page 15 of 19

16 ([ )) i trictly monotonic Since! take only non-negative value one conclude that! 0," an interval 0,! [ ), and! i a emi-algebraic homeomorphim The curve branch w 1 = w!! "1 ha the deired propertie Ω The curve branch w 1 in Lemma 35 i differentiable by Lemma 32 It i claimed that w i differentiable at 0 if the map! of the proof i differentiable at 0 Firt uppoe that! i differentiable Then w = w 1!! i differentiable Converely, if w i differentiable, then "! ( 0) =! ( t) = w( t) " w( 0) # $ %t (with 0 <! ) on ome initial piece of [ 0,! ), which! t implie differentiability of! (cf Lemma 32) Lemma 35 will be applied in the following way: One want to check whether a function f : k n! k i differentiable at ome point z Then one conider all compoition f! w, [ ) " k n i differentiable and w( 0) = z One may aume that w i injective where w : 0,! Then the curve! i alo differentiable Suppoe that the compoition f! w 1 i differentiable Then alo f! w = f! w 1!! i differentiable Thu, to check differentiability of f at z, it uffice to conider differentiable curve w with w t! w( 0) = t Now the ground ha been prepared for a characterization of local differentiability in term of a local Lipchitz condition: THEOREM 36 A continuou emi-algebraic function f : k n! k i differentiable at ome point z!k n if and only if there are a neighborhood U! k n of z and ome 0 <! "k uch that f x! f ( z) " # $ x! z for all x!u PROOF Firt uppoe that the Lipchitz condition hold It i claimed that f! w i differentiable at 0 for all curve branche w : 0,! at 0 By Lemma 35, it uffice to conider curve branche with w t [ ) " k n with w( 0) = z that are differentiable! w( 0) = t Then f! w( t)! f! w( 0) " # $ w( t)! w( 0) = # $t Lemma 32 (or Theorem 31) how that f! w i differentiable at 0 Now uppoe that differentiability hold at z It i claimed that there i ome neighborhood f ( x)! f ( z) U of z uch that i bounded if x varie in U \ { z} Let B be the cloed unit ball x! z in k n, S n!1 the unit phere Define U = z + B and g :U \ { z}! [ 0," ) : x! f x # f ( z) Thi i a continuou emi-algebraic map Moreover, h :[ 0,! ) " [ 0,1) : a " a 1+ a i a monotonic emi-algebraic homeomorphim To how that g i bounded one can prove equivalently that 1!h! g( U \ { z} ) x # z Aume that the claim i fale The graph of h! g, G, i a emi-algebraic et that i contained in k n+1 For each 0 < r < 1 in k, the et B [ r,1] = y!k n r " y " 1 { } i emialgebraically compact Continuity of h! g implie that 1!h! g z + B [ r,1] The aumption page 16 of 19

17 !k n+1!g Curve election yield a continuou emi-algebraic curve [ ) " k n+1 with w( 0) = ( z,1) and w( ( 0,! )) " G The projection k n+1! k n onto [ ) # U i not contant near 0 "!! w( 0) = t for all t![ 0," ) " µ for all [ ) Then h( g!!! w( t) ) " µ implie that z,1 branch w : 0,! the firt n component i denoted by! The curve!! w : 0," By Lemma 35 one can modify the curve o that!! w t Thi i a differentiable curve, hence g!!! w i bounded, ay g!!! w t t! 0," Thu, w t Ω w t =!! w t 1+ µ < 1, and $ µ ' (,h! g!!! w( t) ) "k n # & 0, % 1+ µ ) ( doe not converge to ( z,1) a t tend to 0 Thi contradiction finihe the proof It i now eay to give a firt order tatement that characterize the emi-algebraic function that are globally differentiable: COROLLARY 37 A continuou emi-algebraic function f : k n! k i differentiable if and only if the following tatement hold:! z "k n #$ > 0 #% > 0! x "k n : x & z < $ ' f ( x) & f ( z) ( % ) x & z Ω COROLLARY 38 Let C( k n ) z be the ring of germ of continuou emi-algebraic function at z!k n Then the following condition about a function germ f are equivalent: (a) f i differentiable at z (b) For each differentiable curve branch w : 0,! homomorphim w * :C k n (c) For each homomorphim h :C k n v V : qf ( V )! " V # $ [ ) " k n with w( 0) = z, the correponding into the valuation ideal I 1 z! C 0,0+ map f! f z z! V onto a valuation ring, with valuation ( ( )) " v V h( ( T 1,,T n )! z ) { }, v V h f! f z Ω Up to thi point, the dicuion of differentiable continuou emi-algebraic function ha dealt only with a concrete geometric etting [21] propoe method to extend contruction with poring of emi-algebraic function to contruction with reduced poring (ection 8, 10 and 14) The final reult how that thee method can be applied here, ie, there i a reflector on the category of reduced poring that extend every poring in a functorial way to a poring of differentiable continuou emi-algebraic function The field of real algebraic number i denoted by R 0, and D R 0 n i the ring of differentiable continuou emi-algebraic function that are defined on R 0 n THEOREM 39 There i an H-cloed monoreflector on the category PO/N of reduced poring uch that the reflection of the free object on n generator, n!!, i the ring D R 0 n page 17 of 19

18 PROOF Schwartz and Madden decribe data that are ufficient to determine an H-cloed monoreflector of reduced poring ([21], p 121): The polynomial ring![ T 1,,T n ], partially ordered by the um of quare, i the free reduced poring on n generator The real cloure i the ring of continuou emi-algebraic function R n n 0! R 0 The ring D R 0 lie between [ ] and it real cloure It i partially ordered by the poitive emi-definite function,! "(![ T 1,,T n ]) i an embedding One need to how that! T 1,,T n ie, the incluion D R 0 n the differentiable continuou emi-algebraic function are cloed under compoition, ie, if g 1,,g n : k m! k i a equence of differentiable function then the compoition : k m! k n! k i differentiable a well, and n [ ] " D R 0 f! g 1,,g n i an epimorphim the incluion! n :! T 1,,T n Cloedne under compoition i clear from the dicuion above It i not quite o obviou that the map! n are epimorphim However, a model for the argument that are needed to prove thi can be found in [21], Example 1411 [ ] " C 1 n ( R 0 ) into the ring of continuouly differentiable emi- The incluion! n :! T 1,,T n algebraic function i known to be an epimorphim (loc cit) In connection with [20], Example 24, one conclude that the upport function from the real pectrum of C 1 n R 0 to it prime pectrum i injective Hence [21], Corollary 147 and Theorem 146 how that the incluion! n :C 1 n R 0 epimorphim Ω n " D( R 0 ) i an epimorphim a well Altogether,! n = " n! # n i an The development preented here focu on algebraic decription of one-ided differentiability Of coure, there i alo a geometric interpretation of differentiability, uch a one-ided tangent direction, their geometric meaning and propertie, etc Thee iue are deferred to a later paper Acknowledgement The author gratefully acknowledge partial upport by Deutche Forchunggemeinchaft (project no SCHW 287/17-1 and SCHW 287/18-2) Bibliography [1] J Bochnak, M Cote, M-F Roy, Géométrie algèbrique réelle Springer, Berlin 1987 [2] N Bourbaki, Algèbre Commutative, Chapitre 8 et 9 Maon, Pari 1983 [3] GW Brumfiel, Witt ring and K-theory Rocky Mountain J Math 14, (1984) [4] G Cherlin, MA Dickmann, Real cloed ring II Model theory Annal Pure Applied Logic 25, (1983) [5] G Cherlin, MA Dickmann, Real cloed ring I Reidue ring of ring of continuou function Fund Math 126, (1986) [6] M Cote, M-F Roy, La topologie du pectre reel In: Ordered Field and Real Algebraic Geometry (Ed DW Duboi, T Recio), Contemporary Math vol 8, Amer Math Soc, Providence 1982, pp [7] H Delf, M Knebuch, Semialgebraic Topology over a Real Cloed Field II Bic Theory of Semialgebraic Soace Math Z 178, (1981) [8] H Delf, M Knebuch, Locally emialgebraic pace Lecture Note in Math, vol 1173, Springer, Berlin 1985 [9] AJ Engler, A Pretel, Valued Field Springer-Verlag, Berlin Heidelberg 2005 page 18 of 19

19 [10] A Ficher, Peano-Differentiable Function in o-minimal Structure Diertation, Univerität Paau, Paau 2006 [11] R Gilmer, Multiplicative Ideal Theory Queen' Paper in Pure and Applied Math, vol 12, Queen' Univerity, Kington 1968 [12] R Huber, C Scheiderer, A relative notion of local completene in emialgebraic geometry Arch Math 53, (1989) [13] M Knebuch, C Scheiderer, Einführung in die reelle Algebra Vieweg, Braunchweig 1989 [14] M Knebuch, D Zhang, Convexity, Valuation and Prüfer Extenion in Real Algebra Documenta Math 10, (2005) [16] S Prieß-Crampe, Angeordnete Strukturen Gruppen, Körper, projektive Ebenen Springer-Verlag, Berlin Heidelberg 1983 [15] H Matumura, Commutative Algebra 2 nd Ed Benjamin/Cumming, Reading, Maachuett 1980 [17] N Schwartz, Real Cloed Space Habilitationchrift, München 1984 [18] N Schwartz, Real Cloed Ring In: Algebra and Order (Ed S Wolfentein), Heldermann Verlag, Berlin 1986, pp [19] N Schwartz, The baic theory of real cloed pace Memoir Amer Math Soc No 397, Amer Math Soc, Providence 1989 [20] N Schwartz, Epimorphic extenion and Prüfer extenion of partially ordered ring manucripta mathematica 102, (2000) [21] N Schwartz, JJ Madden, Semi-algebraic Function Ring and Reflector of Partially Ordered Ring Lecture Note in Mathematic, Vol 1712, Springer-Verlag, Berlin 1999 Niel Schwartz, Fakultät für Informatik und Mathematik, Univerität Paau, Potfach 2540, Paau, Germany Schwartz@fimuni-paaude page 19 of 19