Algorithms yield upper bounds in differential algebra
aa r X i v : . [ m a t h . A C ] A p r ALGORITHMS YIELD UPPER BOUNDSIN DIFFERENTIAL ALGEBRA
WEI LI, ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON
Abstract.
Consider an algorithm computing in a differential field with sev-eral commuting derivations such that the only operations it performs withthe elements of the field are arithmetic operations, differentiation, and zerotesting. We show that, if the algorithm is guaranteed to terminate on everyinput, then there is a computable upper bound for the size of the output of thealgorithm in terms of the input. We also generalize this to algorithms workingwith models of good enough theories (including for example, difference fields).We then apply this to differential algebraic geometry to show that thereexists a computable uniform upper bound for the number of components ofany variety defined by a system of polynomial PDEs. We then use this boundto show the existence of a computable uniform upper bound for the eliminationproblem in systems of polynomial PDEs with delays. Introduction
Finding uniform bounds for problems and quantities (e.g., consistency testing orcounting of solutions) is one of the central questions in differential algebra. In [26], itwas demonstrated that, in commutative algebra, one can show the existence of suchbounds as a consequence of theorems about nonstandard extensions of standardalgebraic objects. This approach was successfully applied in the differential algebracontext in [11] and [8, Section 6] for establishing, for example, the existence of auniform bound in the differential Nullstellensatz. Furthermore, in [25], the authorsused methods of proof theory to extract explicit bounds based on nonstandardexistence proofs.The present paper can be viewed as an alternative approach, in which we derivethe existence of a computable uniform bound for an object from the existence of analgorithm for computing the object. More precisely, let T be a complete recursivetheory. The most relevant examples for us would be the theory of differentiallyclosed fields in zero characteristic with m commuting derivations and the theoryof existentially closed difference fields, others include algebraically closed and realclosed fields. Consider an algorithm A performing computations in a model of T that is restricted to using only definable functions when working with elements ofthe model (for formal definition, we refer to Section 4.1) and required to terminatefor every input.We show that there is a computable upper bound for the size of the output of A interms of the input size of A . We apply this to the Rosenfeld-Gr¨obner algorithm [2]that decomposes a solution set of a system of polynomial PDEs into componentsand is such an algorithm. This allows us to show that there is a uniform upper Mathematics Subject Classification.
Primary 12H05, 12H10, 03C10; Secondary 03C60,03D15. bound for the number of components of any differential-algebraic variety definedby a system of polynomial PDEs. We also show how this bound for the number ofcomponents leads to a uniform upper bound for the elimination problem in systemsof polynomial PDEs with delays.A bound for the number of components of varieties defined by polynomial ODEsappeared in [18], as did a bound for the elimination problem for polynomial ODEswith delays. These bounds are based on the application of the Rosenfeld-Gr¨obneralgorithm, which, if applied in this situation to ODEs, outputs equations whoseorder does not exceed the order of the input. This allowed to restrict to a finitelygenerated subring of the ring of differential polynomials and use tools from algebraicgeometry. It is non-trivial to generalize this to polynomial PDEs because the ordersin the output of the Rosenfeld-Gr¨obner can be larger then the orders of the input.Another key ingredient in the ODE case to obtain the bound in [18] was an analysisof differential dimension polynomials. A significant difference of our present PDEcontext with the ordinary case that these polynomials behave less predictably underprojections of varieties (compare [18, Lemma 6.16] and Lemma 6.3). To overcomethis difficulty, we use again our bound for the Rosenfeld-Gr¨obner algorithm.We believe that our method can also be applied to obtain bounds for otheralgorithms in differential algebra such as [1, Algorithm 3.6] and for algorithms fromother theories, e.g. [7, Algorithm 3] for systems of difference equations. Since thereducibility of a polynomial can be expressed as a first-order existential formula, itseems plausible that the same methods could be applied to other algorithms dealingwith difference [5] and differential-difference [6] equations that use factorizationbecause the corresponding theories satisfy the requirements of our approach [14,17, 23]. However, we leave these for future research.The paper is organized as follows. Section 2 contains definitions and notationused in Section 3 to state the main results. Bounds for an algorithm working witha model of a theory T are established in Section 4. These results are applied todifferential algebra in Section 5. Further applications to delay PDEs are given inSection 6. 2. Basic notions and notaiton
Definition 2.1 (Differential-difference rings) . • A ∆- σ -ring ( R , ∆ , σ ) is a commutative ring R endowed with a finite set ∆ = { ∂ , . . . , ∂ m } of commuting derivations of R and an endomorphism σ of R suchthat, for all i , ∂ i σ = σ∂ i . • When R is additionally a field, it is called a ∆- σ -field. • If σ is an automorphism of R , R is called a ∆- σ ∗ -ring. • If σ = id, R is called a ∆-ring or differential ring. • For a commutative ring R , h F i denotes the ideal generated by F ⊂ R in R . • For ∆ = { ∂ , . . . , ∂ m } , let Θ ∆ = { ∂ i · . . . · ∂ i m m | i j > , j m } . • For θ = ∂ i · . . . · ∂ i m m ∈ Θ ∆ , we let ord θ = i + . . . + i m . For a non-negativeinteger B , we denote Θ ∆ ( B ) := { θ ∈ Θ ∆ | ord θ B } . • For a ∆-ring R , the differential ideal generated by F ⊂ R in R is denoted by h F i ( ∞ ) ; for a non-negative integer B , we introduce the following ideal of R : h F i ( B ) := h θ ( F ) | θ ∈ Θ ∆ ( B ) i . LGORITHMS YIELD UPPER BOUNDS IN DIFFERENTIAL ALGEBRA 3
Definition 2.2 (Differential polynomials) . Let R be a ∆-ring. The differentialpolynomial ring over R in y = y , . . . , y n is defined as R{ y } ∆ := R [ θy k | θ ∈ Θ ∆ ; 1 k n ] . The structure of a ∆-ring is defined by ∂ i ( θy k ) := ( ∂ i θ ) y k for every θ ∈ Θ ∆ . Definition 2.3 (Differential-difference polynomials) . Let R be a ∆- σ -ring. Thedifferential-difference polynomial ring over R in y = y , . . . , y n is defined as R [ y ∞ ] := R [ θσ i y k | θ ∈ Θ ∆ ; i >
0; 1 k n ] . The structure of ∆- σ ring is defined by σ ( θσ j y k ) := θσ j +1 y k and ∂ i ( θσ j y k ) :=( ∂ i θ ) σ j y k for every θ ∈ Θ ∆ and j > σ -polynomial is an element of R [ y ∞ ]. Given B ∈ N , let R [ y B ] denote thepolynomial ring R [ θσ j y k | θ ∈ Θ ∆ ( B ); 0 j B ; 1 k n ] . For the notions from logic that we use, see [19, Sections 2.1-2.2].3.
Main results
For clarity, we gather our main results in one section.
Theorem 3.1 (Upper bound for irreducible components for PDEs) . There ex-ists a computable function
Comp( m, n ) such that, for every differential field k with a set of m commuting derivations ∆ and finite F ⊂ k { y , . . . , y n } ∆ with max { ord F, deg F } s , the number of components in the variety defined by F = 0 does not exceed Comp( m, max { n, s } ) . Additional details and proof are given in Theorem 5.11.
Theorem 3.2 (Upper bound for elimination in delay PDEs) . For all non-negativeintegers r , m and s , there exists a computable B = B ( r, m, s ) such that, for all: • non-negative integers q and t , • a ∆ - σ -field k with char k = 0 and | ∆ | = m , • sets of ∆ - σ -polynomials F ⊂ k [ x t , y s ] , where x = x , . . . , x q , y = y , . . . , y r ,and deg y F s ,we have (cid:10) σ i ( F ) | i ∈ Z > (cid:11) ( ∞ ) ∩ k [ x ∞ ] = { }⇐⇒ h σ i ( F ) | i ∈ [0 , B ] (cid:11) ( B ) ∩ k [ x B + t ] = { } . The two preceding theorems are proved using our main technical result aboutalgorithms performing computations in complete recursive theories. Stating it pre-cisely requires defining admissible algorithms carefully, so we postpone it untilSection 4 and give here a simplified and informal version of the statement.
Theorem 3.3 (Algorithm yields a bound, stated precisely as Theorem 4.5) . Thereexists a computable function with input • complete recursive theory T ; • an algorithm A performing computations in a model of T restricted to usingonly definable functions when working with elements of the model; • positive integer ℓ WEI LI, ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON that computes a number N such that for every model M of T and every a ∈ M ℓ the size of the output of A with input a does not exceed N . For the application of this to the Rosenfeld-Gr¨obner algorithm, see Theorem 5.8.4.
Bounds for the output size of algorithms over complete theories
In this section, we will use the formalism of oracle Turing machines [24, § Setup.
To consider an algorithm dealing with elements of a (not necessarilycomputable) model of a theory T , we will “encapsulate” the elements of the modelgiven to the algorithm into an oracle that allows to perform only first-order op-erations with them as defined below. For other approaches that could be used toformalizing computations in arbitrary structures, see [9, §
1] and [4, § Definition 4.1 ( T -oracle) . Let L be a language and T be a theory in L . Forelements a , . . . , a ℓ of a model M of T , any oracle that supports the followingqueries: given a formula ϕ ( x , . . . , x ℓ ), the oracle returns the value ϕ ( a , . . . , a ℓ ) in M , will be denoted by O M ( a , . . . , a ℓ ). Definition 4.2 (Total algorithm over T ) . An oracle Turing machine A will becalled a total algorithm over T if, for all positive integers ℓ , every model M of T and every a , . . . , a ℓ ∈ M , the machine with every input and oracle O M ( a , . . . , a ℓ )is guaranteed to terminate.4.2. Auxiliary bound and result.Lemma 4.3.
There is an algorithm that takes as input: • language L ; • complete recursive theory T given by a Turing machine producing its axioms; • a total algorithm A over T ; • positive integers ℓ and N ; • a string S in the input alphabet of A ;and computes • a first-order formula ϕ = ϕ T, A ( ℓ, S , N ) in L in ℓ variables and • a number N := N T, A ( ℓ, S , N ) such that, for any model M of T and tuple a ∈ M ℓ , the following are equivalent:(1) the sentence ϕ ( a ) is true in M ;(2) algorithm A with input S and oracle O M ( a ) terminates after performing atmost N queries to the oracleand if these statements are true, then the bitsize of the output of A with input S and oracle O M ( a ) does not exceed N .Proof. We describe an algorithm for computing ϕ T, A ( ℓ, S , N ) and N T, A ( ℓ, S , N ).Fix some L , T, A , ℓ , and S .We will describe an algorithm that, for a given positive integer s , computesfirst-order formulas ψ s and q s in L in the variables x = ( x , . . . , x ℓ ) and a positiveinteger N s such that, for every model M of T and every a ∈ T ℓ LGORITHMS YIELD UPPER BOUNDS IN DIFFERENTIAL ALGEBRA 5 • ψ s ( a ) is true in M iff algorithm A with input S and oracle O M ( a ) will performat least s queries; • if ψ s ( a ) is true in M , then the result of the s -th query will be q s ( a ); • if algorithm A with input S and oracle O M ( a ) performs at most s queries, thenthe bitsize of the output does not exceed N s .Fix some s > ψ , . . . , ψ s − , q , . . . , q s − , and N , . . . , N s − . Assume that A with input S has performed s − s -th query will be performed is determined by theresults of the first s − r ∈ { True , False } s − . It will representpossible results of the first s − L : ψ r ( x ) := ψ s − ( x ) ∧ s − ^ i =1 ( q i ( x ) ⇐⇒ r i ) , where we assume ψ = True. The algorithm uses the recursivity and completnessto check whether the sentence ∃ x ψ r ( x ) is false in T [19, Lemma 2.2.8]. If it is,then there is no oracle of the form O M ( a ) such that A will perform at least s − r , . . . , r s − .In the case of ∃ x ψ r ( x ) is true in T , the algorithm will run A with input S and an oracle O r that works as follows. For the first s − O r will return r , . . . , r s − . For all subsequent queries, it always returns True. The algorithm willstop the execution of A if A makes an s -th query to the oracle, and denote theformula in the query by q r .Since ∃ x ψ r ( x ) is true in T , O r gives the same responses to the first s − O M ( a ). Since A must terminate in finite time for everysuch oracle, one of the following must happen:(1) A will perform an s -th query.(2) A will terminate after performing only s − q r to bethe s -th query. In the latter case, the algorithm will define N r to be the bitsize ofthe output. Then the algorithm computes ψ s ( x ) := _ q r is defined ψ r ( x ) , q s ( x ) := ^ q r is defined ( ψ r ( x ) = ⇒ q r ( x )) , N s − := max N s − , X N r is defined N r ! , where we assume N − = −∞ . If the set { r | q r is defined } is empty, the al-gorithm sets ψ s ( x ) = False and q s ( x ) = True. Finally, the algorithm returns ϕ T, A ( ℓ, S , N ) := ¬ ψ N +1 and N T, A ( ℓ, S , N ) := N N . (cid:3) Lemma 4.4.
Let T be a theory and M an ℵ -saturated model. Let U ⊃ U ⊃ U ⊃ . . . be a sequence of definable sets in M n such that ∞ T i =1 U i = ∅ . Then thereexists N such that U N = ∅ .Proof. Assume the contrary, that is, that U i = ∅ for every i >
1. We will showthat ∞ T i =1 U i = ∅ . WEI LI, ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON
We show that a collections of formulas { x ∈ U i } ∞ i =1 is finitely satisfiable. Indeed,let S ⊂ Z > be a finite set and N = max S . Then T i ∈ S U i = U N = ∅ . Due tocompactness, the countable collection { x ∈ U i } ∞ i =1 is satisfiable in some elemen-tary extension of M . Since M is ℵ -saturated, this collection is satisfiable in M .Therefore, ∞ T i =1 U i = ∅ . (cid:3) Main result.Theorem 4.5.
There exists a computable function
Size T, A ( ℓ, r ) with input • complete recursive theory T ; • total algorithm A over T ; • positive integers ℓ and r that computes a number N such that for every model M of T , every a ∈ M ℓ , andevery string S in the alphabet of A of size at most r the bitsize of the output of A with input S and oracle O M ( a ) does not exceed N .Proof. We will describe an algorithm for computing Size T, A ( ℓ, r ). We fix T , A , ℓ ,and r . We will consider S of length at most r and describe how to compute a boundfor the bitsize of the output given that the input is S . Taking the maximum overall S of length at most r (there are finitely many of them), we obtain Size A ,T ( ℓ, r ).The algorithm will compute ϕ i := ϕ T, A ( ℓ, S , i ) for i = 1 , , . . . using the algo-rithm from Lemma 4.3. For each ϕ i , the algorithm will check whether the formulais equivalent to True in T using the recursivity and completeness [19, Lemma 2.2.8].If this is true, the algorithm stops and returns N T, A ( ℓ, S , i ) (see Lemma 4.3).It remains to show that the described procedure terminates in finitely many steps.Let M be an ℵ -saturated model of T (it exists, for example, due to [19, Theo-rem 4.3.12]). For every i = 1 , , . . . , we introduce a definable set U i := { a ∈ M ℓ | ϕ i ( a ) = False } . Notice that U i = ∅ if and only if ( ϕ i ⇐⇒ True) in T . Then the definition of ϕ i ’s implies that U ⊃ U ⊃ . . . . Assume that T ∞ i =1 U i is not empty and choose anelement a in it. Then A will not terminate in finitely many steps with input S andoracle O M ( a ). Thus, T ∞ i =1 U i = ∅ . Lemma 4.4 implies that there exists N suchthat U N = ∅ . Then our algorithm will terminate after considering ϕ N . (cid:3) Applications to differential algebra
In this section, we will apply the results of Section 4 to the theory of differentiallyclosed fields with several commuting derivations.5.1.
Preparation.Notation 5.1.
Let m be a positive integer. • The language of partial differential rings with m commuting derivation is de-noted by L m := { + , − , · , , , ∂ , . . . , ∂ m } . We add a separate functional symbolfor subtraction for convenience. • The theory of partial differentially closed fields with m commuting derivationsof characteristic zero is denoted by DCF m . Recall that DCF m is complete [21,Corollary 3.18] and recursive [21, Section 3.1] (see also [15]). LGORITHMS YIELD UPPER BOUNDS IN DIFFERENTIAL ALGEBRA 7
Notation 5.2.
Let m, n, h be positive integers and k a differential field with a setof m commuting derivations ∆ = { ∂ , . . . , ∂ m } . • Pol k ( m, n, h ) denotes the space of all differential polynomials over k in n vari-ables of order at most h and degree at most h . • The dimension of Pol k ( m, n, h ) (which does not depend on of k ) will be denotedby PolDim( m, n, h ). Notation 5.3.
Let m , ℓ and n be positive integers. • Let L m ( x , . . . , x ℓ ) { y , . . . , y n } ∆ denote the ring of differential polynomials indifferential variables y , . . . , y n with respect to m derivations with the coef-ficients being terms in the language L m in x , . . . , x ℓ (that is, elements of Z { x , . . . , x ℓ } ∆ ).This is a computable differential ring with m commuting derivations. Inwhat follows, we will assume that the algorithms use dense representation tostore these polynomials (that is, store all the coefficients up to certain orderand certain degree). • Let k be a differential field with m derivations and a ∈ k ℓ . Then, for T ∈L m ( x , . . . , x ℓ ) { y , . . . , y n } ∆ , we define T ( a ) ∈ k { y , . . . , y n } ∆ to be the resultof evaluating the coefficients of T at a . Definition 5.4. A differential ranking for k { z , . . . , z n } ∆ is a total order > on Z := { θz i | θ ∈ Θ ∆ , i n } satisfying, for all i , 1 i m : • for all x ∈ Z , ∂ i ( x ) > x and • for all x, y ∈ Z , if x > y , then ∂ i ( x ) > ∂ i ( y ). Notation 5.5.
For a ∆-field k and f ∈ k { z , . . . , z n } ∆ \ k and differential ranking > , • lead( f ) is the element of Z of the highest rank appearing in f . • The leading coefficient of f considered as a polynomial in lead( f ) is denoted byin( f ) and called the initial of f . • The separant of f is ∂f∂ lead( f ) . • The rank of f is rank( f ) = lead( f ) deg lead( f ) f . The ranks are compared first withrespect to lead, and in the case of equality with respect to deg. • For S ⊂ k { z , . . . , z n } ∆ \ k , the set of initials and separants of S is denoted by H S . Definition 5.6 (Characteristic sets) . • For f, g ∈ k { z , . . . , z n } ∆ \ k , f is said to be reduced w.r.t. g if no properderivative of lead( g ) appears in f and deg lead( g ) f < deg lead( g ) g . • A subset
A ⊂ k { z , . . . , z n } ∆ \ k is called autoreduced if, for all p ∈ A , p isreduced w.r.t. every element of A \ { p } . One can show that every autoreducedset is finite [13, Section I.9]. • Let A = A < . . . < A r and B = B < . . . < B s be autoreduced sets ordered bytheir ranks (see Notation 5.5). We say that A < B if – r > s and rank( A i ) = rank( B i ), 1 i s , or – there exists q such that rank( A q ) < rank( B q ) and, for all i , 1 i < q ,rank( A i ) = rank( B i ). • An autoreduced subset of the smallest rank of a differential ideal I ⊂ k { z , . . . , z n } ∆ is called a characteristic set of I . One can show that everynon-zero differential ideal in k { z , . . . , z n } ∆ has a characteristic set. WEI LI, ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON • A radical differential ideal I of k { z , . . . , z n } ∆ is said to be characterizable if I has a characteristic set C such that I = h C i ( ∞ ) : H ∞ C . Proposition 5.7.
There is a computable function that, for a given positive integer m , computes a total algorithm RG m over DCF m such that, for every differentialfield k with m derivations and a ∈ k ℓ , the input-output specification of RG m withoracle O k ( a ) is the following: Input: finite subsets A and S of L m ( x , . . . , x ℓ ) { y , . . . , y n } ∆ ; Output: a list of tuples C , . . . , C N from L m ( x , . . . , x ℓ ) { y , . . . , y n } ∆ such that C ( a ) , . . . , C N ( a ) is the output of the Rosenfeld-Gr¨obner algorithm [2, Theorem 9] with input ( A ( a ) , S ( a )) .Proof. [2, Theorem 9] states that the only operations performed by the Rosenfeld-G¨obner algorithm with the elements of the ground differential field are arithmeticoperations, differentiation, and zero testing. Algorithm RG m is constructed to workexactly in the same way as the Rosenfeld-Gr¨obner algorithm with the only differencethat the elements of the ground differential field will be represented as L ( a ), where L ∈ L m ( x , . . . , x ℓ ) { y , . . . , y n } ∆ . The arithmetic operations and differentiationscan be performed with L , and zero testing can be performed using the oracle, so RG will be able to perform the same computations as the Rosenfeld-Gr¨obner algorithm.Due to [2, Theorem 5], the Rosenfeld-Gr¨obner algorithm is guaranteed to termi-nate on every input. Hence, the same is true for RG m . (cid:3) Bounds.Theorem 5.8 (Upper bound for Rosenfeld-Gr¨obner algorithm) . There exists acomputable function
RG( m, n, ℓ ) such that, for every differential field k with m derivations and subsets A, S ⊂ Pol k ( m, n, n ) with | A | , | S | ℓ , and every differentialranking, the output of the Rosenfeld-Gr¨obner algorithm [2, Theorem 9] on A and S will produce at most RG( m, n, ℓ ) components with all the orders and degrees notexceeding RG( m, n, ℓ ) .Proof. We fix m , n , and ℓ and compute the total algorithm RG m over DCF m from Proposition 5.7. Let a be the set of all the coefficients of A and S . Then | a | N := 2 ℓ PolDim( m, n, n ). The sets A and S can be presented as evaluations ofsubsets e A, e S ⊂ L m ( x , . . . , x N ) { y , . . . , y n } ∆ at a such that the orders and degreesof e A, e S in y , . . . , y n do not exceed n and every coefficient is a single variable x i .Then the size of ( e A, e S ) is bounded by a computable function S ( m, n, N ).We run RG m with the input I = ( e A, e S ) and oracle O ( a ). Lemma 4.5 impliesthat the bitsize of the output will not exceed Size RG m , DCF m ( N, S ( m, N, n )).Since each component takes at least one bit, polynomial of degree d or order d has at least d coefficients (due to the dense representation of the polynomials,see Notation 5.2) requiring at least one bit each, the number of components, thedegrees and orders do not exceed the bitsize of the output. Therefore, we can setRG( m, n, ℓ ) = Size RG , DCF m ( N, S ( m, N, n )). (cid:3) Corollary 5.9.
There exists a computable function
CharSet( m, n, ℓ ) such that,for every computable differential field k with m derivations and subsets A, S ⊂ Pol k ( m, n, n ) with | A | , | S | ℓ , and every differential ranking, the ideal LGORITHMS YIELD UPPER BOUNDS IN DIFFERENTIAL ALGEBRA 9 p h A i ( ∞ ) : S ∞ can be written as an intersection of at most CharSet( m, n, ℓ ) char-acterizable differential ideals defined by their characteristic sets with respect to theranking of order and degree not exceeding CharSet( m, n, ℓ ) .Proof. Theorem 5.8 implies that there exists a representation q h A i ( ∞ ) : S ∞ = ( h C i ( ∞ ) : H C ) ∩ . . . ∩ ( h C N i ( ∞ ) : H C N ) , where H C i is the product of the initials and separants of C i , and C i is the char-acteristic presentation [2, Definition 8] of h C i i ( ∞ ) : H ∞ C i for every 1 i N . Asnoted in [2, p. 108] a characteristic set of h C i i ( ∞ ) : H ∞ C i can be obtained from C i by performing reductions until it will become autoreduced. Since differential re-duction is a part of the Rosenfeld-G¨obner algorithm, it can also be performed by atotal algorithm over DCF m . Therefore, as in the proof of Theorem 5.8, Lemma 4.5implies that h C i i ( ∞ ) : H ∞ C i has a characteristic set with degrees and order boundedby a computable function of the degrees and orders of C i . The latter are boundedby a computable function RG due to Theorem 5.8. Composing these two bounds,we obtain a desired function CharSet( m, n, ℓ ). (cid:3) Lemma 5.10.
There exists a computable function
PrimeComp( m, n ) such thatfor every partial differential field k with m derivations, every ranking, and everycharacterizable differential ideal I defined by a characteristic set C ⊂ Pol k ( m, n, n ) with respect to this ranking, we have(1) the number of prime components of I does not exceed PrimeComp( m, n ) ;(2) every prime component of I has a characteristic set with respect to theranking with orders and degrees bounded by PrimeComp( m, n ) .Proof. Let H be the product of the initials and separants of C . [2, Theorem 4]implies that the number of prime components of h C i ( ∞ ) : H ∞ is equal to the numberof prime components of the algebraic ideal ( h C i ( ∞ ) : H ∞ ) ∩ R n , where R n is thering of differential polynomials of order at most n . Since the degrees of elementsof C are bounded by n , the B´ezout inequality implies that there is a computablebound D for the degree of I ∩ R n in terms of m and n , so this gives a bound forthe number of components.Let P , . . . , P ℓ be the prime components of I . For every 1 i ℓ , P i ∩ R n isa prime algebraic ideal, and its zero set can be defined by equations of degree atmost deg( P i ∩ R n ) due to [12, Proposition 3]. Therefore, for each 2 i ℓ , wecan choose a polynomial in ( P \ P i ) ∩ R n of degree at most deg( P i ∩ R n ). Theirproduct Q has degree at most deg( I ∩ R n ) D . Observe that P = P : Q ∞ ⊂ I : Q ∞ = ( P : Q ∞ ) ∩ . . . ∩ ( P ℓ : Q ∞ ) = P . Thus, applying Corollary 5.9 to a pair (
C, HQ ) and using that | C | PolDim( m, n ),we show that P has a characteristic set with orders and degrees bounded byCharSet( m, D + n, PolDim( m, n )). (cid:3)
Theorem 5.11 (Upper bound for the components of a differential variety and theirnumber) . There exists a computable function
Comp( m, n ) such that, for all non-negative integers m , n and h and a partial differential field k with m derivationsand finite set F ⊂ Pol k ( m, n, h ) :(1) the number of components in the variety defined by F = 0 does not exceed Comp( m, max { n, h } ) ; (2) for every differential ranking and every component X of the variety F = 0 , X has a characteristic set with respect to the ranking with orders and degreesbounded by Comp( m, max { n, h } ) .Proof. Consider any differential ranking. By replacing F with the basis of itslinear span, we will further assume that | F | PolDim( m, n, h ) (see Notation 5.2).Corollary 5.9 implies that p h F i ( ∞ ) can be represented as an intersection of at most N characterizable ideals with characteristic sets C , . . . , C N of order and degree atmost N , where N := CharSet( m, max { n, h } , PolDim( m, n, h )) . Lemma 5.10 applied to each of C , . . . , C N implies that the number of compo-nents of the variety defined by F = 0 does not exceed N · PrimeComp( m, N ),and each of them has a characteristic set with orders and degrees not exceedingPrimeComp( m, N ). (cid:3) Remark.
It was shown in [11, Theorem 6.1] that there exists a (not necessarilycomputable) bound for the degrees and orders a characteristic set of a prime differ-ential ideal. The second part of Theorem 5.11 implies that there is a computablebound. 6.
Application to delay PDEs
In this section, we will show how Theorem 5.11 applies to the problem of elimi-nation of unknowns in delay PDEs.6.1.
Bounds for Kolchin polynomials for algebraic PDEs.Definition 6.1.
We will say that a ∆-variety X ⊂ A n is bounded by N if N > max( n, m ) ( m = | ∆ | ) and X can be defined by equations of order and degree atmost N . Notation 6.2.
For a numeric polynomial ω ( t ) = m P i =0 a i (cid:0) t + ii (cid:1) , we set | ω | := m X i =0 | a i | . Lemma 6.3.
There exists a computable function
KolchinProj( N ) such that forevery • differential variety X ⊂ A n bounded by N , • irreducible component X ⊂ X , • and linear projection π : A n → A ℓ ,we have | ω Y | KolchinProj( N ) , where Y := π ( X ) Kol .Proof.
By performing a linear change of variables, we reduce the problem to thecase in which π is the projection to the first ℓ coordinates. Consider a ranking suchthat • x ℓ + i is greater than every derivative of x j for every i > j ℓ ; • the restriction of the ranking on x , . . . , x ℓ is an orderly ranking. LGORITHMS YIELD UPPER BOUNDS IN DIFFERENTIAL ALGEBRA 11
Theorem 5.11 implies that X has a characteristic set C with respect to this rankingwith the order bounded by a computable function of N . Since a characteristic set of Y can be obtained from C by selecting the polynomials only in the first ℓ variables,there is a charactersitic set of Y with respect to the orderly ranking with theorder bounded by a computable function of N . Then [16, Proposition 3.1] and [16,Fact 2.1] imply that | ω Y | is bounded by a computable function of N . (cid:3) Proposition 6.4.
There exists an algorithm that, for every computable function g ( n ) : Z > → Z > , produces a number Len g such that, for every sequence of Kolchinpolynomials ω > ω > . . . > ω ℓ such that | ω i | < g ( i ) for every i ℓ , we have ℓ < Len g .Proof. By replacing g ( n ) with n + max k n g ( k ), we can further assume that g ( n ) is in-creasing and g ( n ) > n . [22, Definition 2.4.9 and Lemma 2.4.12] define a computableorder-preserving map c from the set of all Kolchin polynomials K to Z m +1 > (consid-ered with respect to the lexicographic ordering). For v = ( v , . . . , v m ) ∈ Z m +1 > , wedefine | v | = v + . . . + v m . For every function g : Z > → Z > , we define e g ( n ) := max ω ∈K , | ω | g ( n ) | c ( ω ) | . Note that if g ( n ) was computable, then e g ( n ) is also computable.The sequence ω > ω > . . . gives rise to a sequence c ( ω ) > lex c ( ω ) > lex . . . in Z m +1 > with | c ( ω i ) | e g ( i ) for every i . [20, Main Lemma] implies that there isan algorithm to compute the maximal length of such a sequence, so there is analgorithm to compute a bound on ℓ from g . (cid:3) Trains of varieties, partial solutions, and their upper bounds.Lemma 6.5.
For every ∆ - σ -field k of characteristic zero, there exists an extension k ⊂ K of ∆ - σ -fields, where K is a differentially closed ∆ - σ ∗ -field.Proof. The proof follows [18, Lemma 6.1] mutatis mutandis and replacing the ref-erence to [3, Theorem 3.15] by [14, Corollary 2.4]. (cid:3)
Notation 6.6.
Within Sections 6.2 and 6.3, we fix a ground ∆- σ field k and adifferentially closed ∆- σ ∗ -field K given by Lemma 6.5 applied to k . All varieties inSections 6.2 and 6.3 are considered over K . Definition 6.7 (Partial solutions) . • For ∆- σ -rings R and R , a homomorphism φ : R −→ R is called a ∆- σ -homomorphism if, for all i , φ∂ i = ∂ i φ and φσ = σφ . • Let R be a ∆- σ -ring containing a ∆- σ -field k . Let k [ y ∞ ] be the ∆- σ -polynomialring over k in y = y , . . . , y r . Given a point a = ( a , . . . , a r ) ∈ R r , there existsa unique ∆- σ -homomorphism over k , φ a : k [ y ∞ ] −→ R with φ a ( y i ) = a i and φ a | k = id . Given f ∈ k [ y ∞ ], a is called a solution of f in R if f ∈ Ker( φ a ). • For a ∆- σ - k -algebra R and I = N or Z , the sequence ring R I has the followingstructure of a ∆- σ -ring (∆- σ ∗ -ring for I = Z ) with σ and ∆ defined by σ (cid:0) ( x i ) i ∈ I (cid:1) := ( x i +1 ) i ∈ I and ∂ j (cid:0) ( x i ) i ∈ I (cid:1) := ( ∂ j ( x i )) i ∈ I . For a k -∆- σ -algebra R , R I can be considered a k -∆- σ -algebra by embedding k into R I in the following way: a ( σ i ( a )) i ∈ I , a ∈ k. For f ∈ k [ y ∞ ], a solution of f with components in R I is called a sequencesolution of f in R . • Given f ∈ R [ y ∞ ], the order of f is defined to be the maximal ord θ + j suchthat θσ j y k effectively appears in f for some k , denoted by ord( f ). • The relative order of f with respect to ∆ (resp. σ ), denoted by ord ∆ ( f ) (resp.ord σ ( f )), is defined as the maximal ord θ (resp. j ) such that θσ j y k effectivelyappears in f for some k . • Let F = { f , . . . , f N } ⊂ k [ y ∞ ], where y = y , . . . , y r , be a set of ∆- σ -polynomials. Suppose h = max { ord σ ( f ) | f ∈ F } . A sequence of tuples( a , . . . , a r ) ∈ K ℓ + h × · · · × K ℓ + h is called a partial solution of F of length ℓ if( a , . . . , a r ) is a ∆-solution of the system in y ∞ ,ℓ + h − : { σ i ( F ) = 0 | i ℓ − } . We associate the following geometric data with the above set F of ∆- σ -polynomials: • the ∆-variety X ⊂ A H defined by f = 0 , . . . , f N = 0 regarded as ∆-equationsin k [ y ∞ ,h ] with H = r ( h + 1), and • two projections π , π : A H −→ A H − r defined by π ( a , . . . , σ h ( a ); . . . ; a r , . . . , σ h ( a r )):= ( a , σ ( a ) , . . . , σ h − ( a ); . . . ; a r , . . . , σ h − ( a r )) ,π ( a , . . . , σ h ( a ); . . . ; a r , . . . , σ h ( a r )):= ( σ ( a ) , . . . , σ h ( a ); . . . ; σ ( a r ) , . . . , σ h ( a r )) . Let σ ( X ) denote the ∆-variety in A H defined by f σ , . . . , f σN , where f σi is theresult by applying σ to the coefficients of f i . Definition 6.8.
A sequence p , . . . , p ℓ ∈ A H is a partial solution of the triple ( X, π , π ) if(1) for all i , 1 i ℓ , we have p i ∈ σ i − ( X ) and(2) for all i , 1 i < ℓ , we have π ( p i +1 ) = π ( p i ).A two-sided infinite sequence with such a property is called a solution of the triple ( X, π , π ). Lemma 6.9.
For every positive integer ℓ , F has a partial solution of length ℓ ifand only if the triple ( X, π , π ) has a partial solution of length ℓ. The system F has a solution in K Z if and only if the triple ( X, π , π ) has a solution.Proof. As in [18, Lemma 6.5]. (cid:3)
Definition 6.10.
For ℓ ∈ N or + ∞ , a sequence of irreducible ∆-subvarieties( Y , . . . , Y ℓ ) in A H is said to be a train of length ℓ in X if(1) for all i , 1 i ℓ , we have Y i ⊆ σ i − ( X ) and(2) for all i , 1 i < ℓ , we have π ( Y i +1 ) Kol = π ( Y i ) Kol . LGORITHMS YIELD UPPER BOUNDS IN DIFFERENTIAL ALGEBRA 13
Lemma 6.11.
For every train ( Y , . . . , Y ℓ ) in X , there exists a partial solution p , . . . , p ℓ of ( X, π , π ) such that for all i , we have p i ∈ Y i . In particular, if thereis an infinite train in X , then there is a solution of the triple ( X, π , π ) .Proof. As in [18, Lemma 6.7]. (cid:3)
For two trains Y = ( Y , . . . , Y ℓ ) and Y ′ = ( Y ′ , . . . , Y ′ ℓ ), denote Y ⊆ Y ′ if Y i ⊆ Y ′ i for each i . Given an increasing chain of trains Y i = ( Y i, , . . . , Y i,ℓ ), (cid:0) ∪ i Y i, , . . . , ∪ i Y i,ℓ Kol (cid:1) is a train in X that is an upper bound for this chain. (For each j , ∪ i Y i,j Kol is anirreducible δ -variety in σ j − (X).) So by Zorn’s lemma, maximal trains of length ℓ always exist in X .For ℓ ∈ N , consider the product X ℓ := X × σ ( X ) × · · · × σ ℓ − ( X )and denote the projection of X ℓ onto σ i − ( X ) by ϕ ℓ,i . Let W ℓ ( X, π , π ) := { p ∈ X ℓ : π ( ϕ ℓ,i ( p )) = π ( ϕ ℓ,i +1 ( p )) , i = 1 , . . . , ℓ − } . Lemma 6.12.
Let ( X, π , π ) be a triple with X bounded by n . Then, for every ℓ ,the number of maximal trains of length ℓ in X does not exceed Comp( m, ℓn ) .Proof. This follow from Theorem 5.11 and rewritten mutatis mutandis for severalcommuting derivations [18, Lemma 6.8]. (cid:3)
Definition 6.13.
Let (
X, π , π ) be a triple and ω ( t ) be a numeric polynomial.We define B ( X, ω ) ∈ Z ∪ {∞} as the smallest value that is greater than the lengthof any train in X with Kolchin polynomials at least ω . Lemma 6.14.
Let X be a differential variety bounded by n such that B ( X, < ∞ .Then B ( X, ω X ) does not exceed the number of components of X plus one.Proof. Denote the number of components in X by N and assume that thereis a train ( Y , . . . , Y N +1 ) with the Kolchin polynomial at least ω X . Theneach of Y , σ − ( Y ) , . . . , σ − N ( Y N +1 ) must be a component of X , so there exist1 i < j N + 1 such that Y j = σ j − i Y i . Thus, there exists an infinitetrain ( Y , . . . , Y i , Y i +1 , . . . , Y j − , σ j − i ( Y i ) , σ j − i ( Y i +1 ) , . . . ) in X . This contradictsto B ( X, < ∞ . (cid:3) Lemma 6.15.
There exists a computable function
Iter( n, D ) such that, for everytriple ( X, π , π ) such that • B ( X, < ∞• X is bounded by n and every numeric polynomial ω ( t ) > , there exists a numeric polynomial ω ( t ) > such that • ω ( t ) < ω ( t ) ; • | ω | Iter( n, B ( X, ω )) ; • B ( X, ω ) Iter( n, B ( X, ω )) .Proof. The proof follows [18, Lemma 6.20]. Let B := B ( X, ω ), and let T be the number of maximal trains of length B in X . We set B := B + T .Lemma 6.12 implies that T is bounded by Comp( m, nB ). Consider the fibered product W B ( X, π , π ), and, for each irreducible components W in it, denote thecorresponding train by Y W . We set (assuming max ∅ = 0) ω := max (cid:8) ω Y W | ω Y W < ω , W is a component of W B (cid:9) . We will show that B ( X, ω ) B + T . Assume that there is a maximal train( Y , . . . , Y B ) in X with the Kolchin polynomial at least ω . Introduce T + 1 trains Z (1) , . . . , Z ( T +1) of length B in X, σ ( X ) , . . . , σ T ( X ), respectively, such that foreach j , Z ( j ) = (cid:0) Z ( j )1 , . . . , Z ( j ) T (cid:1) := ( Y j , . . . , Y j + B − ) . Then for each j , consider a maximal train ˜ Z ( j ) of length B containing Z ( j ) . So σ − j +1 ( ˜ Z ( j ) ) is a maximal train of length B in X . There are two cases to consider:(Case 1) (cid:8) ω Y W ( t ) (cid:12)(cid:12) ω Y W ( t ) < ω ( t ) , W is a component of W B (cid:9) = ∅ . In this case, ˜ Z (1) is a train in X with Kolchin polynomial at least ω . This contra-dicts the definition of B ( X, ω ).(Case 2) (cid:8) ω Y W ( t ) (cid:12)(cid:12) ω Y W ( t ) < ω ( t ) , W is a component of W B (cid:9) = ∅ . By the definition of B ( X, ω ), for every j , ω σ − j +1 ( ˜ Z ( j ) ) ( t ) < ω ( t ). This impliesthat, for each j , ω σ − j +1 ( ˜ Z ( j ) ) ( t ) = ω ( t ) . Since there are only T maximal trains in X of length B , there exist a < b suchthat σ − a +1 ( ˜ Z ( a ) ) = σ − b +1 ( ˜ Z ( b ) ) =: Z. Since ω Z = ω , there exists ℓ such that ω Z ℓ = ω . Since ω σ − a +1 ( Z ( a ) ℓ ) = ω and σ − a +1 ( Z ( a ) ℓ ) ⊆ Z ℓ we have σ − a +1 ( Z ( a ) ℓ ) = Z ℓ . Similarly, we can show σ − b +1 ( Z ( b ) ℓ ) = Z ℓ . Hence, σ − a +1 ( Y a + ℓ − ) = σ − a +1 ( Z ( a ) ℓ ) = σ − b +1 ( Z ( b ) ℓ ) = σ − b +1 ( Y b + ℓ − ) . Thus, we have Y b + ℓ − = σ b − a ( Y a + ℓ − ). This contradicts the fact that B ( X, < ∞ .It remains to show that | ω | is bounded by a computable function of n and B .Let W be a component of W B such that ω Y W = ω . Let Y W = ( Y W, , . . . , Y W,B ).There exists 1 i B such that ω Y i = ω . Since Y i is the Kolchin closure of alinear projection of a component of W B and W B is bounded by B n , Lemma 6.3implies that | ω | is bounded by a computable function of n and B .Taking Iter( n, D ) to be the maximum of the computable bounds for B ( X, ω )and | ω | , we conclude the proof. (cid:3) Definition 6.16.
Let n be a positive integer and ω ( t ) be a numeric polynomialsuch that ω >
0. We define B ( n, ω ) ∈ Z ∪ {∞} as the smallest value such that,for every affine differential variety X bounded by n , if there exists a train in X with Kolchin polynomial at least ω of length at least B ( n, ω ), then there exists aninfinite train in X . Proposition 6.17. B ( n, is bounded by a computable function A ( n ) . LGORITHMS YIELD UPPER BOUNDS IN DIFFERENTIAL ALGEBRA 15
Proof.
We recursively define the following function G ( n ) on nonnegative integers G (0) := max (cid:0) Components( n ) + 1 , KolchinProj( n ) (cid:1) ,G ( j + 1) := Iter( n, G ( j )) , j > . Consider a variety X bounded by n such that there is no infinite train in X , that is B ( X, < ∞ . Lemma 6.14 implies that B ( X, ω X ) − X . Hence Theorem 5.11 implies that B ( X, ω X ) Comp( n ) + 1.Lemma 6.3 implies that | ω X | KolchinProj( n ). Repeatedly applying Lemma 6.15,we obtain a sequence of numeric polynomials ω := ω X > ω > ω > . . . such that, for every 1 i L , we have B ( X, ω i ) G ( i ) and | ω i | G ( i ). Sincethe Kolchin polynomial are well-ordered, there exists L such that ω L = 0. Proposi-tion 6.4 implies that L Len G . Hence, B ( X, G (Len G ), where the right-handside is a computable function of n . Set A ( n ) := G (Len G ), then B ( n, A ( n ). (cid:3) Corollary 6.18.
For all r , m and s ∈ Z > , and a set of ∆ - σ polynomials F ⊂ k [ y s ] with | ∆ | = m , deg F s and | y | = r , F = 0 has a solution in K Z if and only if F = 0 has a partial solution of computable length A (max { r, m, s } ) .Proof. As in [18, Corollary 6.21]. (cid:3)
Upper bound for delay PDEs.Theorem 6.19.
For all non-negative integers r , m , and s , there exists a computable B = B ( r, m, s ) such that, for all: • non-negative integers q and t , • ∆ - σ -fields k with char k = 0 and | ∆ | = m , • sets of ∆ - σ -polynomials F ⊂ k [ x t , y s ] , where x = x , . . . , x q , y = y , . . . , y r , and deg y F s ,we have (cid:10) σ i ( F ) | i ∈ Z > (cid:11) ( ∞ ) ∩ k [ x ∞ ] = { } ⇐⇒ h σ i ( F ) | i ∈ [0 , B ] (cid:11) ( B ) ∩ k [ x B + t ] = { } . Proof.
The proof closely follows [18, Theorem 6.22]. The “ ⇐ = ” implicationis straightforward. We will prove the “ = ⇒ ” implication. For this, let A := A (max { r, m, s } ) from Corollary 6.18, and let B be a computable bound obtainedfrom [10, Theorem 3.4] with m ← m, n ← r ( A + s + 1) , h ← s, and D ← s. By assumption,(1) 1 ∈ (cid:10) σ i ( F ) | i ∈ Z > (cid:11) ( ∞ ) · k ( x ∞ )[ y ∞ ] . Suppose that(2) h σ i ( F ) | i ∈ [0 , A ] (cid:11) ( B ) ∩ k [ x B + t ] = { } . If 1 ∈ (cid:10) σ i ( F ) | i ∈ [0 , A ] (cid:11) ( B ) · k ( x B + t )[ y ∞ ,A + s ] , then there would exist c i,j ∈ k ( x B + t )[ y ∞ ,A + s ] such that(3) 1 = X θ ∈ Θ ∆ ( B ) A X j =0 X f ∈ F c i,j θ ( σ j ( f )) . Multiplying equation (3) by the common denominator in the variables x B + t , weobtain a contradiction with (2). Hence, by [10, Theorem 3.4],1 / ∈ (cid:10) σ i ( F ) | i ∈ [0 , A ] (cid:11) ( ∞ ) · k ( x B + t )[ y ∞ ,A + s ] . By Lemma 6.5, there exists a differentially closed ∆- σ ∗ -field extension L ⊃ k ( x ∞ ) ⊃ k ( x B + t ). Then differential Nullstellensatz implies that the system of differentialequations { σ i ( F ) = 0 | i ∈ [0 , A ] } in the unknowns y ∞ ,A + s has a solution in L . Then the system F = 0 has a partialsolution of length A + 1 in L . Now from (1), we see that the system F = 0 has nosolutions in L Z . Together with the existence of a partial solution of length A + 1,this contradicts to Corollary 6.18. (cid:3) Acknowledgments
This work was partially supported by the NSF grants CCF-1564132, CCF-1563942, DMS-1760448, DMS-1760413, DMS-1853650, DMS-1853482, and DMS-1800492; and the NSFC grants (11971029,11688101).
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URL https://doi.org/10.1007/BF01388493 . KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,No.55 Zhongguancun East Road, 100190, Beijing, China
E-mail address : [email protected] CUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd, Queens, NY11367, USA and CUNY Graduate Center, Mathematics and Computer Science, 365 FifthAvenue, New York, NY 10016, USA
E-mail address : [email protected] ´Ecole Polytechnique, Institut Polytechnique de Paris, LIX, CNRS, 1 rue Honor´ed’Estienne d’Orves, 91120, Palaiseau, France and National Research University HigherSchool of Economics, Moscow, Russia E-mail address : [email protected] University of California, Berkeley, Department of Mathematics, Berkeley, CA94720-3840, USA
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