Janet's approach to presentations and resolutions for polynomials ...

12 W. Plesken, D. Robertz2) In addition, assign values to the standard basis of R q , i. e. define θ : {e 1 ,...,e q } → Zand extend θ to Mon(R q ) by θ(me i ) = d(m) + θ(e i ) for all m ∈ Mon(R),i = 1,...,q.Then(R q ) ≤i := ⊕ (R q ) j with (R q ) j := ⊕Kmj≤im∈Mon(R q )θ(m)=jturns R q into a filtered R-module: {0} =: (R q ) ≤i0 −1 ⊂ (R q ) ≤i0 ⊆ (R q ) ≤i0 +1 ⊆ ... ⊆ R q =∪ i (R q ) ≤i with R ≤i (R q ) ≤j ⊆ (R q ) ≤i+j for all i ≥ 0,j ≥ i 0 . Moreover, gr(R q ) is isomorphicas graded module to the free gr(R)-module gr(R) q .The proof is straightforward. A (θ-)compatible term order < on Mon(R q ) is defined byexactly the same conditions as in Definition 25. Now Theorem 26 remains valid for θ-compatible term orders rather than term orders compatible with the grading given there.Finally, Corollary 27 also remains true and the proofs need just minor adjustments.Now follows the application to solution theory of linear pdes. Since we want to countpower series solutions, the differential field K needs further specification: Assume thatK is the field of fractions of some field of (real or complex) analytic functions in theindependent variables x 1 ,...,x n and let ∂ i be the partial derivatives with respect to x ifor i = 1,...,n.Let S be a left R-module, whose elements we usually think of as functions in x 1 ,...,x n ,on which ∂ i acts as partial derivative with respect to x i , though the module is arbitraryat this stage. Then each p = p(∂) ∈ R q can be viewed as a row in R 1×q acting on thecolumns in S q×1 by matrix multiplication. Since columns in S q×1 can also be consideredas R-module homomorphisms in Hom R (R 1×q , S), the well known algebraic interpretationof a solution u ∈ S q×1 of the linear pde-system p(∂)u = 0 with p(∂) ∈ G for some finitesubset G ⊂ R q is simply that of a homomorphism in Hom R (R q /X, S) with X := 〈G〉 R .Unfortunately, this algebraic way of viewing solutions is rather restricted. To get to aninterpretation of the (generalized) Hilbert series, we introduce the following notion forpoints with coordinates in the field of constants K 0 of K:Definition 29. A point (a 1 ,...,a n ) ∈ K n 0 is called regular for a (finite) subset G ⊂ R q ,if there is a K 0 -subalgebra A of K closed under the partial derivatives ∂ i satisfying1) G ⊂ A〈∂ 1 ,...,∂ n 〉 q ,2) the Janet basis of 〈G〉 R can be obtained by computation over A〈∂ 1 ,...,∂ n 〉 only,3) (a 1 ,...,a n ) is not a pole for any of the elements of A.Clearly, the set of regular points is Zariski dense in K0 n ; in fact, for reasonably well behavedK, it can be easily obtained by keeping track of the denominators occurring in G andduring the run of Janet’s algorithm. For these regular points, the Hilbert series counts thefree Taylor coefficients for power series solutions. To set up notation, let (a 1 ,...,a n ) ∈ K0nbe a regular point for G ⊂ R q , and let S := K 0 [[x 1 − a 1 ,...,x n − a n ]] be the ringof formal power series over the field K 0 in the x i − a i . Note, S is an A〈∂ 1 ,...,∂ n 〉-module for a suitable A, as just defined. The elements of K Mon(Rq )0 , i. e. the mappingsf : Mon(R q ) → K 0 , are in bijection to S q×1 :⎛⎞K Mon(Rq )0 → S q×1 : f ↦→ s =⎝ ∑i∈(Z ≥0 ) n f(∂ i e 1 )(x − a)ii!,...,∑i∈(Z ≥0 ) n f(∂ i e q )(x − a)ii!where (x − a) i := (x 1 − a 1 ) i1 · · · (x n − a n ) in and i! := i 1 ! · · · i n !. Let X := 〈G〉 R . Theassociated multiple-closed set of leading monomials of its non-zero elements gives rise to⎠tr,