JASS 07 - Polynomals: Ther Power and How to Use Them Dfferental Polynomals Stephan Rtscher March 18, 2007 Abstract Ths artcle gves an bref ntroducton nto dfferental polynomals, deals and manfolds and ther correlatons. Some examples for bad behavour (n comparson to algebrac polynomals) are gven. 1
Contents 1 Algebrac Aspects 3 1.1 Defntons....................................... 3 1.2 Nonrecursve Ideals.................................. 3 1.3 Reducton....................................... 4 2 Geometrc Ascpects 7 2.1 Manfolds....................................... 7 2.2 Algebrac Representaton.............................. 7 3 Concluson 8 2
1 Algebrac Aspects 1.1 Defntons Defnton 1.1 (Dfferental Rng) A dfferental rng R s a rng wth dfferental operators = {δ 1,..., δ m } and for all, j: δ (ab) = (δ a)b + a(δ b) δ (a + b) = δ a + δ b δ δ j = δ j δ Θ = s called the free abelan monod of dervatons. Example 1.2 Let R be an arbtrary rng and δ(x) = 0 for x R. Then R s a dfferental rng wth = {δ}. Consder the polynomals over a rng R wth varables θx for θ Θ and {1,..., n}. They form an dfferental rng denoted by R{X} = R{x 1,..., x n }. The degree deg(f) for a monomal f = s =1 vα s defned as n the algebrac case: deg(f) = s =1 α where v = θ x wth θ Θ, {1,..., n} On the other hand one defnes the order of a varable v to be the number of dfferentatons contaned n v, so ord(δ α x) = n =1 α wth multndex α. To combne these two, one defnes the weght wt(f) = r =1 β ord(v ). Defnton 1.3 (Dfferental Ideal) An dfferental deal I s a deal of R wth δ : δi I. The dfferental deal generated by a set G s denoted by [G]. Example 1.4 (Dfferental Ideal) The followng polynomals are members of the dfferental deal I generated by x 2 over F {x} wth = {d} (x (k) := d k x): (You obtan them by dfferentatng x 2 (x p for the general case) and then cancellng terms by lnear combnatons.) 1. fx 2 for f F {x} 2. fx (1) x 3. f(x (2) x + (x (1) ) 2 ) and therefore f(x (1) ) 2 x 4. f(2x (1) x (2) x + (x (1) ) 3 ) and therefore f(x (1) ) 3 5. f(x (k) ) s for some s > 1 6.... 1.2 Nonrecursve Ideals Example 1.5 Consder over Z{x} wth = {d} the functons f = (d x) 2 for 0 and I k = [f 0,..., f k ] Then I clam: I 0 I 1... Proof Frst note that deg(f ) = 2, wt(f ) = 2. If we dfferentate a monomal, all resultng terms have the same degree as the orgnal monomal. Therefore d j f s homogeneous of degree 2. The weght of the terms ncreases by one per dfferentaton and therefore d j f s sobarc of weght 2 + j. k j=0 α,jd j (f ). Now assume f n I n 1, ths means f n = n 1 =0 If deg(α,j ) 1 for some, j, these terms must cancel because deg(f n ) = 2 and the dervatves of f are homogeneous. So we can assume that α,j Z. Analogously we can assume α,j = 0 for j 2n 2 because wt(f n ) = 2n and d j f are sobarc 3
of weght 2 + j. So the equaton smplfes to f n = c 0 d 2n f 0 + c 1 d (2n 2) f 1 +... + c n 1 d 2 f n 1 for c Z. d 2n f 0 contans the monomal x (2n) x. No other term contans an x (that s not dervated). So c 0 must be zero. For analogous reasons also c must be 0. But f n 0, so we have a contradcton. Example 1.6 Let S N 0 and I S = [{f : S}]. Then f I S S. (Ths follows from a proof smlar to the one above.) So for a nonrecursve set S N 0 there s no algorthm to decde f a gven dfferental polynomal g s n I S. Ths means that we have to consder nce deals f we want to do calculaons, e.g. recursvely generated or even fntely generated deals. 1.3 Reducton Defnton 1.7 (Rankng) Let < be a total orderng on the set ΘX of dfferental varables whch fulflls the followng propertes: v < w θv < θw for all v, w ΘX, θ Θ v θv for v ΘX Then < s called rankng of ΘX. Now let be X = {x 1,..., x n } wth x 1 <... < x n. Example 1.8 (Lexcographc Rankng on ΘX) Consder a monomal orderng < on the dfferental operators Θ. Then the lexcographc orderng s gven by θx < ηx k ff < k or = k and θ < η. Example 1.9 (Dervaton Rankng on ΘX) Consder a monomal orderng < on the dfferental operators Θ. Then the dervaton orderng s gven by θx < ηx k ff θ < η or θ = η and < k. For X = = 1 there s only one rankng: x () < x (+1) Defnton 1.10 (Admssble Orderng) Let < be a total orderng on the set M of monomals of F {X}. Then < s called admssble ff 1. The restrcton of < to ΘX s a rankng. 2. 1 f for all f M 3. f < g hf < hg for all f, g, h M Let be f = r =1 vα wth v 1 >... > v r and g = s =1 wβ wth w 1 >... > w s. Example 1.11 (Lexcographc Orderng on M) Gven an rankng on ΘX. f < lex g ff k r, s : v = w for < k and v k < w k or v k = w k and α < β or v = w for r and r < s. Example 1.12 (Graded (by Degree) Reverse Lexcographc Orderng on M) Gven an rankng on ΘX. f < degrevlex g ff deg(f) < deg(g) or deg(f) = deg(g) and f < revlex g. 4
Defnton 1.13 Let f R{X} be an dfferental polynomal and fx a monomal orderng. Then lm(f) denotes the leadng monomal of f wth respect to the monomal orderng. lc(f) denotes the leadng coeffcent of f and lt(f) = lc(f)lm(f) the leadng term of f. Defnton 1.14 f s reduced by g to h ff θ Θ, m M such that lm(f) = mlm(θg) and h = f lc(f) lc(g) mθg. f s reducable by g, ff there s an h such that f s reduced by g to h. Procedure : Reduce ( f, g, r = 1) f ( deg ( lm ( f ) ) < deg ( lm ( g ) ) wt ( lm ( f ) ) < wt ( lm ( g ) ) ) return f ; f ( lm ( g ) lm ( f ) ) return Reduce ( f ( l t ( f )/ l t ( g ) ) g, g ) ; f o r ( = r ; <= m; ++) { t = Reduce ( f, d e l t a ( g, ), ) ; f ( t!= f ) return Reduce ( t, g ) ; } return f ; Ths procedure termnates. On every recursve call ether f s reduced and therefore the leadng monomal gets smaller or g s dervated (delta(g, )) and therefore the weght of g ncreases. So after a fnte number of calls Reduce termnates. The returned polynomal cannot be reduced by g further because n Reduce the recucton wth respect to all dervatves of g (whch have no bgger weght or degree than f) s tred. Ths process can - as n the algebrac case - be generalzed to a reducton by several polynomals, but n general the remander of the reducton s dependent on the order of these polynomals. Defnton 1.15 (Monodeal) E M s called a monodeal ff ME E and lm( E) E. Please note that n contrast to the algebrac case the defnton of the monodeal needs an monomal orderng and s hghly dependend on ths (as we wll see n the examples). Defnton 1.16 (Standard Bass) G I s called a standard bass ff lm(g) generates lm(i) as monodeal. We now wll nvestgate the monodeals generated by the polynomal x 2, for whch we already consdered the dfferental deal. Example 1.17 (Monodeal - Lexcographc Orderng) The followng monomals are members of the monodeal I generated by x 2 over F {x} wth = {d} usng lexcographc orderng (x (k) := d k x): 1. mx 2 for m M 2. mx (1) x 3. mx (2) x 4. mx (k) x 5. BUT (x (k) ) r / I Example 1.18 (Monodeal - Graded Reverse Lexcographc Orderng) The followng monomals are members of the monodeal I generated by x 2 over F {x} wth = { } usng graded lexcographc orderng (x (k) := d k x): 5
1. mx 2 for m M 2. mx (1) x 3. m(x (1) ) 2 4. mx (1) x (2) 5. m(x (2) ) 2 6. mx (k) x (k+1) 7. mx 2 (k) Theorem 1 Let G be a set of polynomals, I a dfferental deal. Then the followng propostons are equvalent: 1. G s a standard bass of I. 2. For f F {X} yelds: f I f s reduced to 0 by G. Proof Let 0 f I. Then f s reducble by G because the leadng monomal of f s n the monodeal generated by I and therefore also n the monodeal generated by G. The reducton of f s h I. Therefore h s reducble agan untl h = 0. The process termnates because the leadng monomal of the polynomal gets smaller n each reducton. Let g G. Then obvously g s reduced to 0 by G and therefore G I. Let f I. Then f s reduced to 0 by G by defnton and therefore lmi lm(mθlm(g)) (otherwse reducton would fal). Example 1.19 Remember I = [x 2 ] over F {x} wth = { }. Then for every r 0 there s an q > 1 such that (x (r) ) q I. LEX: lm(d( r =1 vα )) = d(v 1 )v α 1 1 r 1 =2 vα f v 1 >... > v r. Therefore (x (r) ) s for every r 0 for some s > 0 s n every standard bass ( nfnte). DEGREVLEX: x 2 s a standard bass. Example 1.20 Conjecture: There s no fnte standard bass for [xx ] for no monomal orderng. Lemma 1.21 The famles of monomals 1. x r x (r) for r 1 2. x tr (r) for r 1 and some t r r + 2 3. x 2 (r) x2 (r+2) x (r+2k r) for r 0 and some k r 4. x 2 (r) x2 (r+3) x (r+3l r) for r 0 and some l r 2r 1 belong to the deal [xx ]. Ths lemma (wthout proof) mples that all mentoned famles of monomals have to be n the monodeal generated by the standard bass. 6
2 Geometrc Ascpects 2.1 Manfolds We choose e.g. F as set of all meromorphc functon. Defnton 2.1 Let Σ be a system of dfferental polynomals over F {x 1,..., x n }, F 1 an extenson of F. If Y = (y 1,..., y n ) F1 n such that for all f Σ f(y 1,..., y n ) = 0, then Y s a zero of Σ. The set of all zeros of Σ (for all possble extentons of F ) s called manfold. Let M 1, M 2 be the manfolds of Σ 1, Σ 2. If M 1 M 2 then M 1 M 2 s the manfold of Σ 1 + Σ 2. M 1 M 2 s the manfold of {AB : A Σ 1, B Σ 2 }. M s called reducble f t s unon of two manfolds M 1, M 2 M. Otherwse t s called rreducble. Lemma 2.2 M s rreducble (AB vanshes over M A or B vanshes over M) Proof Assume A, B such that AB vanshes over M, but A, B don t. Then the manfolds of Σ + A, Σ + B are proper parts of M, ther unon s M. Let M be proper unon of M 1, M 2 wth systems Σ 1, Σ 2. Then A Σ be dfferental polynomals that do not vansh over M. A 1 A 2 vanshes over M. Theorem 2 Every manfold s the unon of a fnte number of rreducble manfolds. Consder dfferental polynomals over F {x} wth = {d} and F the meromorphc functons: Example 2.3 Let Σ = {f} wth f = x 2 (1) 4x. Then df = 2x (1)(x (2) 2). x (1) = 0 has the soluton x(t) = c. Lookng at f, only c = 0 s vald. x (2) 2 = 0 has the soluton x(t) = (x + b) 2 + c. Agan c = 0. There are no other solutons. 2.2 Algebrac Representaton Theorem 3 Let Σ = [f 1,..., f k ] wth manfold M. If g vanshes over M then g s Σ for some s N 0. So the manfolds are represented by perfect deals. Theorem 4 Every perfect dfferental deal has a fnte bass. Let Σ be a fnte system of dfferental polynomals. Queston: Is f Σ? Resolve Σ nto prme deals. f must be member of each of these prme deals. Test f the remander of f wth respect to the characterstc sets of the prme deals s zero. 7
3 Concluson We have seen that dfferental polynomals can be used to model dfferental equaton systems. There are many problems n contrast to algebrac polynomals. E.g. there are dfferental deals that have no fnte (even no recursve) bass and there are fntely generated deals that have (presumably) no fnte standard bass. The dffcultes that arse when tryng to fnd standard bases are also caused by the fact that dfferental monomal deals depend on orderng. We saw that manfolds (solutons of dfferental equaton systems) correspond to perfect deals, that are easer to handle than general dfferental deals. To conclude we remember that for some mportant problems fnte algorthms exst, e.g. for the reducton wth respect to a fnte bass and the membershp test for perfect deals. References [GMO94] G. Gallo, B. Mshra, and F. Ollver. Some constructons n rngs of dfferental polynomals. Lecture Notes In Computer Scence, 539, 1994. [Oll90] F. Ollver. Standard bases of dfferental deals. Lecture Notes In Computer Scence, 508, 1990. [Rt50] J. Rtt. Dfferental Algebra. AMS, 1950. [Zob06] A. I. Zobnn. On standard bases n rngs of dfferental polynomals. Journal of Mathematcal Scences, 135(5), 2006. 8