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Identifying Complete Differential Varieties Identifying complete differential varieties William Simmons Identifying complete differential varieties Algebraic varieties and completeness William Simmons δ-varieties and δ- completeness University of Illinois at Chicago How do we identify δ-complete Kolchin Seminar in Differential Algebra varieties? New York City, NY The valuative criterion in March 9, 2012 action Where to from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties Conventions and terminology Identifying complete In the differential setting, our fields are all DCF0. differential varieties Otherwise they are algebraically closed. William Our varieties V ; W ;::: , are subsets of affine or projective Simmons space defined by appropriate polynomial equations. Algebraic All Kolchin-closed subsets (δ-varieties) we consider are varieties and completeness defined over a fixed arbitrary DCF0 denoted by F. We δ-varieties and δ- denote by K any DCF0 containing F. completeness A finite-rank δ-variety is one that has dimension < ! for How do we identify any/all of the usual ordinal-valued ranks on DCF0. δ-complete varieties? Let R be a δ-ring, K ⊇ R a δ-field, L a δ-field and The valuative ' : R ! L a δ-homomorphism . We say R is a maximal criterion in action δ-ring (with respect to K) if ' does not extend to a Where to δ-homomorphism with domain contained in K but strictly from here? containing R and codomain a δ-field. William Simmons University of Illinois at Chicago Identifying complete differential varieties Compactness Identifying complete differential varieties A topological space X is compact if every open cover of X William Simmons admits a finite subcover. Algebraic varieties and Example: Closed, bounded intervals of (by the l.u.b. completeness R δ-varieties axiom). and δ- completeness Compact subsets of Hausdorff spaces are closed, and being How do we compact is most valuable when a space is Hausdorff. identify δ-complete Fact: Algebraic varieties are compact (algebraic geometers varieties? The valuative say quasicompact), but generally not Hausdorff. criterion in action Where to from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties Completeness and basic properties Identifying complete differential Definition varieties An algebraic variety V is complete if for every variety W the William Simmons projection π2 : V × W ! W is a closed map (takes closed sets to closed sets). Algebraic varieties and completeness δ-varieties Completeness is a local property. and δ- completeness Closed subsets of complete varieties are complete. How do we identify Products of complete varieties are complete. δ-complete varieties? If ' : V ! W is a morphism of varieties and V is The valuative complete, then '(V ) is complete. criterion in action Morphisms of complete irreducible varieties into affine Where to from here? space are constant. William Simmons University of Illinois at Chicago Identifying complete differential varieties The fundamental theorem of elimination theory Identifying complete differential Example varieties William Infinite affine varieties are not complete: Consider the second Simmons projection of Z := xy − 1 = 0. The image π2(Z) = fy 6= 0g, 1 Algebraic which is not closed in A . varieties and completeness n Fact: a complete subset of P must be projectively closed. In δ-varieties and δ- fact, completeness How do we Theorem identify δ-complete Projective varieties are complete. varieties? The valuative criterion in action Proof. Where to Projective Nullstellensatz + linear algebra. from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties δ-completeness Identifying complete δ-completeness of an affine or projective δ-variety is defined by differential varieties the obvious generalization. Most properties carry over to the William differential case: Simmons δ-completeness is a local property. Algebraic varieties and completeness δ-complete varieties are projectively closed. δ-varieties Kolchin-closed subsets of δ-complete varieties are and δ- completeness δ-complete. How do we identify Products of δ-complete varieties are δ-complete. δ-complete varieties? If ' : V ! W is a morphism of δ-varieties and V is The valuative δ-complete, then '(V ) is δ-complete. criterion in action BUT morphisms of irreducible δ-complete varieties into Where to from here? affine space need not be constant. William Simmons University of Illinois at Chicago Identifying complete differential varieties Pong Identifying 1 complete Kolchin's example: P is not complete. differential varieties William Theorem Simmons (Pong) Only finite-rank δ-varieties can be δ-complete. Algebraic varieties and completeness Theorem δ-varieties and δ- (Pong) A δ-complete variety is isomorphic to a δ-complete completeness variety contained in 1. How do we A identify δ-complete 1 varieties? So it suffices to consider subsets of A ; for convenience we The valuative 1 criterion in usually work with P . action Where to Freitag has generalized much of Pong's paper to the case of from here? several commuting derivations (i.e., DCF0;m). William Simmons University of Illinois at Chicago Identifying complete differential varieties Positive quantifier elimination Identifying complete differential Positive formulas are those that contain no negation symbols. varieties William Theorem Simmons (van den Dries) Let T be an L-theory and '(¯x) an L-formula. Algebraic varieties and Then ' is equivalent modulo T to a positive quantifier-free completeness formula iff for all M; N j= T , substructures A ⊆ M, ¯a 2 A, δ-varieties and δ- and L-homomorphisms f : A !N we have completeness M j= '(¯a) =) N j= '(f (¯a)). How do we identify δ-complete varieties? Proof. The valuative 0 criterion in Slick choice of auxiliary theory T + positive atomic diagram + action compactness. Where to from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties (Part of) a valuative criterion Identifying Let p = (p : p ) 2 V ⊆ 1, and let F ⊆ R ⊆ K be a maximal complete 0 1 P differential δ-ring. We say p is in R if either p0=p1 or p1=p0 is. varieties William Simmons Theorem (Pong, S) Let V be a δ-subvariety of 1. If for every p 2 V Algebraic P varieties and and maximal δ-ring R we have p 2 R, then V is δ-complete. completeness δ-varieties and δ- completeness Proof. How do we Use van den Dries' PQE to show the formula '(x0; x1; y¯) identify δ-complete varieties? (9x0; x1)((^i Pi (x0; x1; y¯) = 0)^ The valuative criterion in action (^j Qj (x0; x1) = 0) ^ (x0 = 1 _ x1 = 1)) Where to from here? is equivalent to a positive quantifier-free formula. William Simmons University of Illinois at Chicago Identifying complete differential varieties Ritt and P. Blum Identifying complete differential varieties Definition William Simmons An element a of a δ-field K is monic over a δ-ring A ⊆ K if a satisfies a δ-polynomial equation xn + f (x) = 0, where Algebraic varieties and f (x) 2 Afxg and n > total degree of terms in f . completeness δ-varieties and δ- Facts: completeness How do we Derivatives of monic elements are monic. identify δ-complete Maximal δ-rings are local differential rings. varieties? m m The valuative If (R; ) is maximal, a is monic over R iff 1=a 2= . criterion in action a 2 K n R iff 1 2 mfag: Where to from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties Morrison Identifying complete differential varieties Theorem William Maximal δ-rings are integrally closed. Simmons Algebraic varieties and Theorem completeness Let (R; m) be a maximal δ-ring. If a satisfies a linear δ-varieties and δ- differential equation over R and 1=a 2= R, then a 2 R. completeness How do we identify Proof. δ-complete varieties? Show a is integral over R. This requires monicness of a; a0;::: The valuative criterion in and order-reducing substitutions given by the linear differential action equation. Where to from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties First fruits Identifying complete differential varieties Proposition William 1 Simmons (S) The projective closure in P of a linear δ-variety is Algebraic δ-complete. varieties and completeness δ-varieties Proof. and δ- completeness Morrison's result + the valuative criterion. How do we identify δ-complete Corollary varieties? (n) The valuative When using the valuative criterion, we may suppose a 6= 0 criterion in action for all n 2 N. Where to from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties Some good examples Identifying complete differential varieties Proposition William 1 Simmons (S) The projective closures in P of the following are δ-complete: Algebraic varieties and 0 completeness Q(x)x = P(x), where Q; P are ordinary polynomials δ-varieties x00 = x3 and δ- completeness xx00 = x0 How do we identify δ-complete varieties? Proof. The valuative Valuative criterion + integral closedness of R + a lot of pencil criterion in action lead. Where to from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties Known to be δ-complete Identifying complete differential General classes: varieties William Linear: all Simmons x(n) = P(x(n−1)) Algebraic (n) varieties and Equations algebraic in F[x ] completeness δ-varieties First-order: and δ- 0 completeness Q(x)x = P(x) How do we 0 n identify (x ) = x + α δ-complete varieties? Second-order: The valuative 00 3 criterion in x = x (and friends) action xx00 = x0 Where to from here? William Simmons University of Illinois at Chicago Identifying complete differential varieties Where to from here? Identifying complete Hope/Guess differential varieties All finite-rank, projectively closed δ-varieties are δ-complete. William Simmons Algebraic Refine 1-preserving algorithms, understand integrality in varieties and completeness this setting better δ-varieties Generalize proof of algebraic completeness or show this and δ- completeness cannot be done How do we identify Direct approach: analyze QE, use Rosenfeld-Groebner δ-complete varieties? algorithm, etc.
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