Modeling Physics with Differential-Algebraic Equations Lecture 5 Differential Algebra COMASIC (M2) Khalil Ghorbal [email protected] K. Ghorbal (INRIA) 1 COMASIC M2 1 / 29 Outline 1 Differential Algebra 2 Multi-Mode DAEs (Structural Analysis) K. Ghorbal (INRIA) 1 COMASIC M2 1 / 29 Summary • Gr¨obner Bases are central objects in Computer Algebra Systems • Elimination Theory generalizes Gaussian elimination (BLT Forms) • Qualitative Analysis of ODE via their algebraic invariant sets This Lecture • Differential Algebra (Quick Introduction) • Hybrid Aspects: Challenges K. Ghorbal (INRIA) 2 COMASIC M2 2 / 29 Generalities R denotes a commutative unitary ring. • a 2 R is a zero divisor if and only if there exists b 2 R, b 6= 0 such that ab = 0. • 0 is a trivial zero divisor. • Domains are rings where the only zero divisor is zero. • Quotient of a ring R by an ideal I : R=I . • f¯ (or f ) is zero in R=I if and only if f is in the ideal I . • R=I , residue class ring (not necessarily a domain). • M is a multiplicatively closed subset of R if and only if m1m2 2 M for all m1, m2 in M. K. Ghorbal (INRIA) 3 COMASIC M2 3 / 29 Prime and Primary Ideals The zero divisors of the residue class ring are important. Prime Ideals The ideal p is said to be prime if and only if • R=p is a domain. • R=p does not have nontrivial zero divisors. • ab 2 p if and only if either a 2 p or b 2 p. Primary Ideals The ideal q is said to be primary if and only if • all zero divisors of R=q are nilpotent. • ab 2 q if and only if either am 2 q or bm 2 q for some positive natural number m. p The radical of a primary ideal, q, is a prime ideal. K. Ghorbal (INRIA) 4 COMASIC M2 4 / 29 Examples • Any maximal ideal is prime, the converse is not true. • For a ring R,(X ) is a prime ideal of R[X ] but it is not maximal ((X ) ⊂ (X ; Y )). • Intuition: points of a given affine space are not the only irreducible varieties { this will be given a precise meaning later. Lines and circles are also irreducible. K. Ghorbal (INRIA) 5 COMASIC M2 5 / 29 Localization −1 • One constructs the ring RM (or M R), the ring R localized at M, r with elements of the form m where r 2 R and s 2 M. • For r1 , r2 in R , the \+" composition law is defined by m2r1+m1r2 . m1 m2 M m1m2 • Let 'M denote the ring homomorphism R ! RM , then −1 'M [('M (I ))] = I : M K. Ghorbal (INRIA) 6 COMASIC M2 6 / 29 Saturation Ideals Let I be an ideal of R and M a multiplicatively closed subset of R. Then I : M = ff 2 R j 9m 2 M ; mf 2 I g : is a Saturation ideal. Let q be a primary ideal. • If M \ I 6= ; then I : M = R. • If M \ q = ; then q : M = q. s Suppose one has a primary decomposition of I : I = \i=1qi , then \ I : M = qi qi \M=; K. Ghorbal (INRIA) 7 COMASIC M2 7 / 29 Sylvester Matrix d d−1 • f = ad X + ad−1X + ··· + a1X + a0 e e−1 • g = be X + be−1X + ··· + b1X + b0 • (d; e) 6= (0; 0) The Sylvester Matrix of f and g is the following (d + e) square matrix. 0 ad 0 ··· 0 be 0 ··· 0 1 Bad−1 ad ··· 0 be−1 be ··· 0 C B C B .. .. C Bad−2 ad−1 . 0 be−2 be−1 . 0 C B . C B . .. .. C B . ad . be C B . C B . C S(f ; g) = B . ··· ad−1 . ··· be−1C B . C B a a ··· a b b ··· . C B 0 1 d−2 0 1 C B . C B 0 a .. 0 b .. C B 0 0 C B . .. .. C @ . a1 . b1 A 0 0 ··· a0 0 0 ··· b0 K. Ghorbal (INRIA) 8 COMASIC M2 8 / 29 Examples f = X 4 + 3X 3 − 1 f = X 2 − X + 1 g = −1 g = X − 2 0−1 0 0 0 1 0 1 1 0 1 0 −1 0 0 −1 −2 1 B C @ A B 0 0 −1 0 C 1 0 −2 @ A 0 0 0 −1 K. Ghorbal (INRIA) 9 COMASIC M2 9 / 29 Resultant R is a domain. Let f and g be two polynomials in R[X ]. The resultant of f and g, res(f ; g), is the determinant of the Sylvester matrix of f and g. Properties • f and g have a common root (on an algebraic field extension of R) if and only if res(f ; g) = 0. • res(f ; g) = (f ; g). That is, there exists polynomials u and v of respective degrees less that e and d, such that res(f ; g) = uf + vg. K. Ghorbal (INRIA) 10 COMASIC M2 10 / 29 Differential Ring Derivation A derivation δ on R is a map R ! R satisfying: • δ(a + b) = δ(a) + δ(b) • δ(ab) = δ(a)b + aδ(b) • Notation: δa is often used instead of δ(a). • δ1 and δ2 commute if and only if δ1δ2a = δ2δ1a for all a 2 R. • High-order derivatives δh are inductively defined as δ(δh−1a). • Ordinary differential ring has a single derivation. • Partial differential ring has a family ∆ = fδ1; : : : ; δmg of pairwise commuting derivations. • Same definitions apply for a field k. K. Ghorbal (INRIA) 11 COMASIC M2 11 / 29 Definitions c 2 R is a constant if and only if δc = 0 for all δ 2 ∆. If k if a differential field, then the set of constants is a subfield of k. Differential Ideal A differential ideal a of a ∆-ring R is an ideal of R such that 8δ 2 ∆; 8a 2 a; δa 2 a : • The intersection of an arbitrary number of differential ideals is a differential ideal • The finite sum of differential ideals is also a differential ideal • [S] denotes the differential ideal generated by S ⊂ R. K. Ghorbal (INRIA) 12 COMASIC M2 12 / 29 Examples Example 1 d • R = Z[X ], ∆ = dX , is a differential ring. • a = (2; 2X ) is a differential ideal. • [X ] = R (δX = 1) Example 2 X d • R = Z[X ; e ], ∆ = dX , is a differential ring. • a = (eX ) is a differential ideal. • [X ] = R (δX = 1) K. Ghorbal (INRIA) 13 COMASIC M2 13 / 29 Derivation Order • Θ: free multiplicative monoid generated by ∆ = fδ1; : : : ; δmg. e1 em • θ 2 Θ has the form δ1 ··· δm , ei 2 N. • order of θ is defined as e1 + ··· + em. e1 em • Θ acts on R by θa = δ1 ··· δm a K. Ghorbal (INRIA) 14 COMASIC M2 14 / 29 Differential Polynomial Ring (1/2) • Let R be a ∆-ring. • ΘX = fXθ;j g1≤j≤n,θ2Θ, n > 0, family of indeterminates. • R[ΘX ] has a unique ∆-ring structure extending the ∆-ring structure of R by δXθ;j = Xδθ;j , for all δ 2 ∆, θ 2 Θ. • R[ΘX ], equipped with this structure is called Differential Polynomial Ring, • in the differential indeterminates Xj = X1;j . • R[ΘX ] is denoted by RfX1;:::; Xng or simply RfX g • We write Xθ;j by θYj (partial derivative of Yj ) • The order of θYj is defined as the order of θ K. Ghorbal (INRIA) 15 COMASIC M2 15 / 29 Differential Polynomial Ring (2/2) • A Differential Monomial is a finite power product of derivatives of the form θYj Q ek • k (θk Yjk ) , the θk (and likewise the Yjk ) are not necessarily distinct • A differential polynomial is a finite sum of terms aM where a 2 R and M is a differential monomial n o • RfX g has a structure of differential ring with ∆ = @ @(θYj ) 1≤j≤n,θ2Θ K. Ghorbal (INRIA) 16 COMASIC M2 16 / 29 Examples • RfX1; X2g, ∆ = fδ1; δ2g 4 3 • θ1 = δ1δ2 4 • θ2 = δ2 3 • θ3 = δ1δ2 2 • (θ1X1)(θ2X1)(θ3X2) X2 is a monomial or order 7 and degree 5 The Wronskian determinant of dimension 2: X1 X2 W = δX1 δX2 is a differential polynomial in RfX1; X2g. • W = X1δX2 − X2δX1 2 2 • δW = X1δ X2 − X2δ X1 • The partial derivative of W with respect to δX2 is X1 K. Ghorbal (INRIA) 17 COMASIC M2 17 / 29 Differential Equations • Given a ∆ differential ring R, to each element f of RfX g, one associates a (partial) differential equation f = 0. • Likewise for a finite subset of S ⊂ RfX g, one gets a system of partial differential equations. • The differential ideal [S] is associated with S and contains all the equations derived from S by addition, multiplication by elements of RfX g and differentiation. K. Ghorbal (INRIA) 18 COMASIC M2 18 / 29 Triangular Forms As seen in the purely algebraic case, one wants to define a weaker notion of triangular forms suitable for the differential case. We thus need to define reduction and hence ordering over differential monomials. Triangular Form A system S ⊂ RfX g is in triangular form if its element can be rearranged as S1; S2;:::; Sk ;::: such that each Sk involves at least one derivative θk Xjk which does not appear in S1;:::; Sk−1. In particular S1 2= R. Example 2 S1 : δ X2 + X2 2 2 S := S2 : δ X2 + X2 + X3 2 S3 : δ X2 + X1 2 S is in triangular form with respect to X2; X3; X1 (or δ X2; X3; X1) There are other definitions of triangular forms. K. Ghorbal (INRIA) 19 COMASIC M2 19 / 29 Ranking Definition A ranking of X1;:::; Xn is a total ordering on ΘX such that for all u; v 2 ΘX and δ 2 ∆, u ≤ δu, and u ≤ v implies δu ≤ δv.