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-1 1 }{CSTYLE "_cstyle12" -1 217 "Courie r" 1 10 0 0 255 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle13" -1 218 "Cou rier" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle14" -1 219 "Tim es" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle9" -1 209 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 201 "" 0 "" {TEXT 206 99 "Basic concepts in constructive differential algebra and their representation in the diffalg package" }}{SECT 0 {PARA 202 "" 0 "" {TEXT 207 71 "Motivation to study differe ntial equations from an algebraic standpoint" }}{PARA 203 "" 0 "" {TEXT 208 65 "Consider the differential system consisting of the two e quations:" }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 68 "eq1 := diff (u(x),x)^2+u(x)^2*diff(V(x),x)^2+2*u(x)*V(x)+2*u(x)^2 =0;" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,**$-%%diffG6$-%\"uG6#%\"xGF.\"\"# \"\"\"*&F+F/-F)6$-%\"VGF-F.F/F0*&F+F0F4F0F/*$F+F/F/\"\"!" }{TEXT 210 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 46 "eq2 := diff(u(x ),x)*diff(V(x),x)-u(x)-V(x) =0;" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>% $eq2G/,(*&-%%diffG6$-%\"uG6#%\"xGF.\"\"\"-F)6$-%\"VGF-F.F/F/F+!\"\"F2F 4\"\"!" }{TEXT 210 1 " " }}}{PARA 206 "" 0 "" {TEXT 211 10 "If a pair \+ " }{TEXT 212 9 "U(x),V(x)" }{TEXT 211 61 " of meromorphic functions is a solution of this system, then " }{TEXT 212 9 "U(x),V(x)" }{TEXT 211 22 " is also a solution of" }}{PARA 206 "" 0 "" {TEXT 211 58 "- th e equations obtained by differentiation, for instance:" }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 12 "diff(eq2,x);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#/,**&-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F,\"\"#\"\"\"-F' 6$-%\"VGF+F,F1F1*&-F'6$F)F,F1-F'6$F4F-F1F1F7!\"\"F2F;\"\"!" }{TEXT 210 1 " " }}}{PARA 206 "" 0 "" {TEXT 211 61 "- the equations obtained \+ by linear combination, for instance:" }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 23 "expand(eq1+2*u(x)*eq2);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#/,(*$-%%diffG6$-%\"uG6#%\"xGF,\"\"#\"\"\"*&F)F--F'6$-% \"VGF+F,F-F.*(F)F.F&F.F0F.F-\"\"!" }{TEXT 210 1 " " }}}{PARA 206 "" 0 "" {TEXT 211 50 "Note that this latter equation can also be written" } }{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 23 "factor(eq1+2*u(x)*eq2); " }}{PARA 205 "" 1 "" {XPPMATH 20 "6#/*$,&-%%diffG6$-%\"uG6#%\"xGF,\" \"\"*&F)F--F'6$-%\"VGF+F,F-F-\"\"#\"\"!" }{TEXT 210 1 " " }}}{PARA 206 "" 0 "" {TEXT 211 10 "Therefore " }{TEXT 212 9 "U(x),V(x)" }{TEXT 211 23 " must be a solution of " }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 16 "op(1,lhs(%)) =0;" }}{PARA 205 "" 1 "" {XPPMATH 20 " 6#/,&-%%diffG6$-%\"uG6#%\"xGF+\"\"\"*&F(F,-F&6$-%\"VGF*F+F,F,\"\"!" } {TEXT 210 1 " " }}}{PARA 203 "" 0 "" {TEXT 208 78 "More generally, con sider a differential system given by equations of the form " }{TEXT 213 15 "p1=0,...., pr=0" }{TEXT 208 11 " where the " }{TEXT 213 2 "pi" }{TEXT 208 45 "'s are polynomials in some unknown functions " }{TEXT 213 10 "u1,..., un" }{TEXT 208 4 " of " }{TEXT 213 12 "x1, ...., xm" } {TEXT 208 36 " and their partial derivatives. The " }{TEXT 213 2 "pi" }{TEXT 208 38 " are called differential polynomials. " }}{PARA 206 "" 0 "" {TEXT 211 3 "If " }{TEXT 212 1 "n" }{TEXT 211 7 "-uplet " }{TEXT 212 16 "U1(x),..., Un(x)" }{TEXT 211 100 " forms a meromorphic solutio n of the system, then it is also a solution of any equation obtained b y:" }}{PARA 206 "" 0 "" {TEXT 211 18 "- differentiating " }{TEXT 212 11 "p1,...., pr" }}{PARA 206 "" 0 "" {TEXT 211 31 "- taking linear com bination of " }{TEXT 212 11 "p1,...., pr" }{TEXT 211 1 " " }}{PARA 206 "" 0 "" {TEXT 211 47 "and therefore taking any linear combination \+ of " }{TEXT 212 11 "p1,...., pr" }{TEXT 211 23 " and their derivatives ." }}{PARA 206 "" 0 "" {TEXT 211 38 "The set of all linear combination s of " }{TEXT 212 11 "p1,...., pr" }{TEXT 211 62 " and their derivativ es is the differential ideal generated by " }{TEXT 212 11 "p1,...., pr " }{TEXT 211 1 "." }}{PARA 206 "" 0 "" {TEXT 211 36 "Now, if some diff erential equations " }{TEXT 212 3 "p=0" }{TEXT 211 17 " in the unknown s " }{TEXT 212 10 "u1,..., un" }{TEXT 211 14 " is such that " }{TEXT 212 3 "p^2" }{TEXT 211 4 " or " }{TEXT 212 3 "p^3" }{TEXT 211 17 " or \+ any power of " }{TEXT 212 1 "p" }{TEXT 211 47 " can be written as a li near combination of the " }{TEXT 212 11 "p1,...., pr" }{TEXT 211 6 " t hen " }{TEXT 212 10 "U1,..., Un" }{TEXT 211 28 " must also be a soluti on of " }{TEXT 212 3 "p=0" }{TEXT 211 1 "." }}{PARA 206 "" 0 "" {TEXT 211 19 "The set of all the " }{TEXT 212 1 "p" }{TEXT 211 22 " such tha t a power of " }{TEXT 212 1 "p" }{TEXT 211 43 " is in the differential ideal generated by " }{TEXT 212 11 "p1,...., pr" }{TEXT 211 55 " is c alled the radical differential ideal generated by " }{TEXT 212 11 "p1, ...., pr" }{TEXT 211 1 "." }}{PARA 206 "" 0 "" {TEXT 211 54 "Actually, the radical differential ideal generated by " }{TEXT 212 11 "p1,...., pr" }{TEXT 211 93 " provides the biggest set of equations having the \+ same meromorphic solutions then the system " }{TEXT 212 15 "p1=0,...., pr=0" }{TEXT 211 1 "." }}{PARA 206 "" 0 "" {TEXT 211 24 "Furthermore, the system " }{TEXT 212 15 "p1=0,...., pr=0" }{TEXT 211 98 " has at l east one meromorphic solution if and only if the radical differential \+ ideal generated by " }{TEXT 212 11 "p1,...., pr" }{TEXT 211 66 " does \+ not contain an element independent of the unknown functions." }}{PARA 203 "" 0 "" {TEXT 208 233 "These two last facts are the basic fundamen tal results of differential algebra. This theory was initiated around \+ 1930 by Ritt for ordinary differential equations and later developed f or partial differential equations by his students." }}{PARA 206 "" 0 " " {TEXT 211 69 "The purpose of constructive differential algebra, and \+ of the package " }{TEXT 212 7 "diffalg" }{TEXT 211 217 ", is to give g ood representations of the radical differential ideals generated by fi nitely many differential polynomial. Doing so we give a good decomposi tion of the solution set of the associated differential system." }} {PARA 203 "" 0 "" {TEXT 208 80 "This help page attempts to give an acc ount of the concepts put in action in the " }{TEXT 213 7 "diffalg" } {TEXT 208 45 " package. For more information, refer to the " } {HYPERLNK 214 "bibliography" 2 "" "references" }{TEXT 208 1 "." }}} {SECT 0 {PARA 202 "" 0 "differential_ring" {TEXT 207 29 "Differential \+ rings and fields" }}{PARA 203 "" 0 "" {TEXT 208 47 "A derivation on a \+ ring is an inner application " }{TEXT 213 1 "d" }{TEXT 208 61 " which \+ is an additive morphism(1) satisfying Leibniz rule(2)." }}{PARA 206 "" 0 "" {TEXT 211 4 "(1) " }{TEXT 212 18 "d(a+b) = d(a)+d(b)" }}{PARA 206 "" 0 "" {TEXT 211 4 "(2) " }{TEXT 212 22 "d(a*b) = d(a)*b+a*d(b)" }}{PARA 203 "" 0 "" {TEXT 208 122 "A differential ring (field) is a co mmutative ring (field) endowed with a finite set of derivations which \+ commute pairwise." }}{PARA 206 "" 0 "" {TEXT 211 10 "Examples: " }} {PARA 206 "" 0 "" {TEXT 211 32 "- The field of rational numbers " } {TEXT 212 1 "Q" }{TEXT 211 62 " endowed with the trivial derivation th at maps any element to " }{TEXT 212 1 "0" }{TEXT 211 40 " is a differe ntial field (of constants)." }}{PARA 206 "" 0 "" {TEXT 211 33 "- The u nivariate polynomial ring " }{TEXT 212 4 "Q[x]" }{TEXT 211 29 " endowe d with the derivation " }{TEXT 212 4 "d/dx" }{TEXT 211 40 " that exten ds the trivial derivation on " }{TEXT 212 1 "Q" }{TEXT 211 15 " and su ch that " }{TEXT 212 10 "d( x ) = 1" }{TEXT 211 44 " is a differential ring. For any polynomial " }{TEXT 212 1 "p" }{TEXT 211 4 " in " } {TEXT 212 4 "Q[x]" }{TEXT 211 2 ", " }{TEXT 212 27 "p =an*x^n + ....+ \+ a1*x + a0" }{TEXT 211 10 ", we have " }{TEXT 212 28 "d(p) = n*an*x^(n- 1)+... + a1" }{TEXT 211 2 ". " }}{PARA 206 "" 0 "" {TEXT 211 52 "- The set of meromorphic functions in two variables " }{TEXT 212 1 "x" } {TEXT 211 5 " and " }{TEXT 212 1 "y" }{TEXT 211 35 " endowed with the \+ usual derivation " }{TEXT 212 4 "d/dx" }{TEXT 211 5 " and " }{TEXT 212 4 "d/dy" }{TEXT 211 14 " according to " }{TEXT 212 1 "x" }{TEXT 211 5 " and " }{TEXT 212 1 "y" }{TEXT 211 25 " is a differential field ." }}{PARA 203 "" 0 "" {TEXT 208 168 "A derivation operator is the com position of a finite number of derivations. The number of derivations \+ involved is the order of the derivation operator (it can be zero)." }} }{SECT 0 {PARA 202 "" 0 "d_polynomial_ring" {TEXT 207 28 "Differential polynomial ring" }}{PARA 203 "" 0 "" {TEXT 208 4 "Let " }{TEXT 213 1 "K" }{TEXT 208 81 " be a differential field of constants of characteri stic zero. All the element of " }{TEXT 213 1 "K" }{TEXT 208 15 " are m apped to " }{TEXT 213 1 "0" }{TEXT 208 25 " by the derivations. Let " }{TEXT 213 1 "F" }{TEXT 208 71 " be the differential field of rational functions in the indeterminates " }{TEXT 213 11 "x1, ..., xm" }{TEXT 208 22 " with coefficients in " }{TEXT 213 1 "K" }{TEXT 208 2 ". " } {TEXT 213 18 "F = K(x1, ..., xm)" }{TEXT 208 39 " is naturally endowed with derivations " }{TEXT 213 11 "d1, ..., dm" }{TEXT 208 31 " accord ing to these variables (" }{TEXT 213 13 "di = d / d xi" }{TEXT 208 2 " )." }}{PARA 206 "" 0 "" {TEXT 211 50 "Given a finite set of differenti al indeterminates " }{TEXT 212 10 "u1,..., un" }{TEXT 211 24 ", we con struct the ring " }{TEXT 212 19 "R = F\{u1, ..., un\} " }{TEXT 211 49 " of differential polynomials. The derivations of " }{TEXT 212 1 "R" } {TEXT 211 27 " extend the derivations on " }{TEXT 212 1 "F" }{TEXT 211 13 " and map the " }{TEXT 212 2 "ui" }{TEXT 211 29 " on their form al derivatives " }{TEXT 212 7 "dj (ui)" }{TEXT 211 1 "." }}{PARA 206 " " 0 "" {TEXT 211 12 "Set-wisely, " }{TEXT 212 1 "R" }{TEXT 211 50 " is the ring of polynomials in the indeterminates " }{TEXT 212 2 "ui" } {TEXT 211 54 " together with all their derivatives up to any order. " }}{PARA 206 "" 0 "" {TEXT 211 19 "For instance, take " }{TEXT 212 8 "F = Q(x)" }{TEXT 211 29 " endowed with the derivation " }{TEXT 212 1 "d " }{TEXT 211 14 " according to " }{TEXT 212 1 "x" }{TEXT 211 50 " and \+ consider a single differential indeterminate " }{TEXT 212 1 "u" } {TEXT 211 28 ". A differential polynomial " }{TEXT 212 1 "p" }{TEXT 211 4 " of " }{TEXT 212 11 "R = Q(x)\{u\}" }{TEXT 211 47 " represents \+ the ordinary differential equation " }{TEXT 212 3 "p=0" }{TEXT 211 23 " with unknown function " }{TEXT 212 4 "u(x)" }{TEXT 211 1 "." }} {PARA 203 "" 0 "" {TEXT 208 10 "Along the " }{TEXT 213 7 "diffalg" } {TEXT 208 10 " package, " }{TEXT 213 11 "x1, ..., xm" }{TEXT 208 34 " \+ are called derivation variables, " }{TEXT 213 1 "K" }{TEXT 208 23 " is referred to as the " }{TEXT 213 18 "field of constants" }{TEXT 208 5 " and " }{TEXT 213 16 "F=K(x1, ..., xm)" }{TEXT 208 23 " is referred t o as the " }{TEXT 213 12 "ground field" }{TEXT 208 4 " of " }{TEXT 213 14 "R=F\{u1,...,un\}" }{TEXT 208 1 "." }}{PARA 203 "" 0 "" {TEXT 208 32 "The differential polynomials of " }{TEXT 213 1 "R" }{TEXT 208 38 " can be denoted using using the MAPLE " }{TEXT 213 4 "diff" } {TEXT 208 5 " and " }{TEXT 213 4 "Diff" }{TEXT 208 33 " functions or t he more compact a " }{TEXT 213 3 "jet" }{TEXT 208 10 " notation:" }} {PARA 206 "" 0 "" {TEXT 212 14 "u[x]^2 - 4*u[]" }{TEXT 211 10 " denote s: " }{TEXT 212 23 "diff(u(x),x)^2 - 4*u(x)" }{TEXT 211 4 " in " } {TEXT 212 7 "Q(x)\{u\}" }}{PARA 206 "" 0 "" {TEXT 212 22 "(1/y)*u[x,x, y] + y + 1" }{TEXT 211 10 " denotes: " }{TEXT 212 32 "(1/y)*diff(u(x,y ),x,x,y) + y + 1" }{TEXT 211 4 " in " }{TEXT 212 9 "Q(x,y)\{u\}" }}} {SECT 0 {PARA 202 "" 0 "ranking" {TEXT 207 7 "Ranking" }}{PARA 203 "" 0 "" {TEXT 208 97 "A ranking is a total order over the set of the deri vatives of the differential indeterminates of " }{TEXT 213 1 "R" } {TEXT 208 31 " that satisfies the two axioms:" }}{PARA 206 "" 0 "" {TEXT 211 4 "(a) " }{TEXT 212 10 "f < dj (f)" }{TEXT 211 20 " for any \+ derivative " }{TEXT 212 1 "f" }{TEXT 211 21 " and each derivation " } {TEXT 212 2 "dj" }{TEXT 211 1 "." }}{PARA 206 "" 0 "" {TEXT 211 4 "(b) " }{TEXT 212 10 "( f <= g )" }{TEXT 211 9 " implies " }{TEXT 212 20 " ( dj (f) <= dj (g) )" }{TEXT 211 21 " for all derivatives " }{TEXT 212 4 "f, g" }{TEXT 211 21 ", and any derivation " }{TEXT 212 2 "dj" } {TEXT 211 1 "." }}{PARA 203 "" 0 "" {TEXT 208 192 "Rankings are the an alogues of term orderings used in Groebner bases algorithms. However, \+ rankings order the derivatives of the differential indeterminates whil e term orderings order monomials." }}{PARA 203 "" 0 "" {TEXT 208 67 "A n elimination ranking between the two differential indeterminates " } {TEXT 213 4 "u, v" }{TEXT 208 6 " (say " }{TEXT 213 5 "u < v" }{TEXT 208 12 ") satisfies " }{TEXT 213 17 "phi (u) < psi (v)" }{TEXT 208 30 " for all derivation operators " }{TEXT 213 8 "phi, psi" }{TEXT 208 1 "." }}{PARA 203 "" 0 "" {TEXT 208 54 "An orderly ranking on the differ ential indeterminates " }{TEXT 213 11 "u1, ..., un" }{TEXT 208 11 " sa tisfies " }{TEXT 213 19 "phi (ui) < psi (uj)" }{TEXT 208 30 " for any \+ derivation operators " }{TEXT 213 8 "phi, psi" }{TEXT 208 24 " such th at the order of " }{TEXT 213 3 "psi" }{TEXT 208 30 " is greater than t he order of " }{TEXT 213 3 "phi" }{TEXT 208 1 "." }}{PARA 203 "" 0 "" {TEXT 208 64 "Given a ranking, we define for every differential polyno mial of " }{TEXT 213 5 "R \\ F" }{TEXT 208 6 ", the " }{HYPERLNK 214 " leader,rank, initial" 2 "diffalg[leader]" "" }{TEXT 208 5 " and " } {HYPERLNK 214 "seprant" 2 "diffalg[leader]" "" }{TEXT 208 1 "." }} {PARA 206 "" 0 "" {TEXT 211 34 "Axioms (a) and (b) imply that, if " } {TEXT 212 1 "v" }{TEXT 211 18 " is the leader of " }{TEXT 212 1 "p" } {TEXT 211 5 " and " }{TEXT 212 3 "phi" }{TEXT 211 41 " is any proper d erivation operator, then " }{TEXT 212 7 "phi (v)" }{TEXT 211 18 " is t he leader of " }{TEXT 212 7 "phi (p)" }{TEXT 211 20 " and the initial \+ of " }{TEXT 212 7 "phi (p)" }{TEXT 211 20 " is the separant of " } {TEXT 212 1 "p" }{TEXT 211 2 ". " }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 207 56 "Representation of differential ring and rankings in the " } {TEXT 215 7 "diffalg" }{TEXT 207 9 " package." }}{PARA 203 "" 0 "" {TEXT 208 2 "A " }{TEXT 213 28 "differential polynomial ring" }{TEXT 208 71 " indicates the data structure (a MAPLE table) returned by the \+ function " }{HYPERLNK 214 "differential_ring" 2 "diffalg[differential_ ring]" "" }{TEXT 208 1 "." }}{PARA 206 "" 0 "" {TEXT 211 127 "This str ucture corresponds to a differential polynomial ring (in the mathemati cal sense) endowed with a ranking and a notation." }}{PARA 203 "" 0 "" {TEXT 208 7 "In the " }{TEXT 213 7 "diffalg" }{TEXT 208 10 " package, " }{TEXT 213 1 "K" }{TEXT 208 179 " is by default the field of ration al numbers. Nonetheless, transcendental and (differential) algebraic e xtensions of the field of rational numbers can be defined with the com mand " }{HYPERLNK 214 "field_extension" 2 "diffalg[field_extension]" " " }{TEXT 208 1 "." }}{PARA 203 "" 0 "" {TEXT 208 44 "In the following \+ example we define the ring " }{TEXT 213 1 "R" }{TEXT 208 64 " of diffe rential polynomials in the differential indeterminates " }{TEXT 213 5 "u,v,w" }{TEXT 208 22 " with coefficients in " }{TEXT 213 6 "Q(x,y)" } {TEXT 208 51 ". The derivations are the derivations according to " } {TEXT 213 1 "x" }{TEXT 208 5 " and " }{TEXT 213 1 "y" }{TEXT 208 28 ". We set an orderly ranking." }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 14 "with(diffalg):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 61 "R := differential_ring(ranking=[[u,v,w]], derivations=[x,y]);" }} {PARA 205 "" 1 "" {XPPMATH 20 "6#>%\"RG%)PDE_ringG" }{TEXT 210 1 " " } }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 11 "indices(R);" }}{PARA 207 "" 1 "" {XPPMATH 20 "657#%(RankingG7%%%LessG%\"wG%\"vG7#%/Indeterm inatesG7#%)NotationG7%F&F'F'7%F&F(F'7%F&F(F(7%F&%\"uGF17#%3Field_of_co nstantsG7#%1To_external_formG7%F&F1F(7%F&F(F17#%5Order_of_derivationsG 7#%3From_external_formG7%F&F'F17#%,DerivationsG7#%-Ground_fieldG7%F&F1 F'7#%%TypeG" }{TEXT 216 1 " " }}}{PARA 206 "" 0 "" {TEXT 211 86 "A con cise presentation of the ranking you have defined can be obtained by t he command " }{HYPERLNK 214 "print_ranking" 2 "diffalg[print_ranking]" "" }{TEXT 211 1 "." }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 17 "p rint_ranking(R);" }}{PARA 208 "" 1 "" {TEXT 217 60 "In lists, leftmost elements are greater than rightmost ones." }{TEXT 217 52 "\nThe deriv atives of [u, v, w] are ordered by grlexA:" }{TEXT 217 25 "\n_U [tau] \+ > _V [phi] when" }{TEXT 217 25 "\n |tau| > |phi| or" }{TEXT 217 71 "\n |tau| = |phi| and _U > _V w.r.t. the list of indeter minates or" }{TEXT 217 62 "\n |tau| = |phi| and _U = _V and tau > phi w.r.t. [x, y]" }}}}{SECT 0 {PARA 202 "" 0 "radical_d_ideal" {TEXT 207 27 "Radical differential ideals" }}{PARA 206 "" 0 "" {TEXT 218 9 "In short:" }{TEXT 211 5 " let " }{TEXT 212 1 "S" }{TEXT 211 5 " and " }{TEXT 212 1 "H" }{TEXT 211 82 " be sets of differential polyno mials. The radical differential ideal generated by " }{TEXT 212 1 "S" }{TEXT 211 8 ", noted " }{TEXT 212 3 "\{S\}" }{TEXT 211 32 ", correspo nds to the the system " }{TEXT 212 3 "S=0" }{TEXT 211 20 ". The satura tion by " }{TEXT 212 1 "H" }{TEXT 211 4 " of " }{TEXT 212 3 "\{S\}" } {TEXT 211 8 ", noted " }{TEXT 212 14 "\{S\}:H^infinity" }{TEXT 211 41 ", corresponds to the differential system " }{TEXT 212 9 "S=0, H<>0" } {TEXT 211 1 "." }}{PARA 203 "" 0 "" {TEXT 208 180 "We will introduce t he concepts of differential ideal and radical differential ideal. The \+ reason for doing so is the following differential analogue of the Hilb ert theorem of zeros." }}{PARA 206 "" 0 "" {TEXT 211 12 "Given a set " }{TEXT 212 1 "S" }{TEXT 211 56 " of differential polynomials, a diffe rential polynomial " }{TEXT 212 1 "p" }{TEXT 211 30 " vanishes on all \+ the zeros of " }{TEXT 212 1 "S" }{TEXT 211 16 " if and only if " } {TEXT 212 1 "p" }{TEXT 211 56 " belongs to the radical differential id eal generated by " }{TEXT 212 1 "S" }{TEXT 211 292 ". This establishes a one-to-one correspondence between the zero sets of differential sys tems and radical differential ideals. Studying and decomposing the zer o set of a differential system amounts to study and decompose the radi cal differential ideal it generates. It is the point of view of " } {TEXT 212 7 "diffalg" }{TEXT 211 1 "." }}{PARA 203 "" 0 "" {TEXT 208 44 "A differential ideal of a differential ring " }{TEXT 213 1 "R" } {TEXT 208 16 " is an ideal of " }{TEXT 213 1 "R" }{TEXT 208 26 " stabl e under derivations." }}{PARA 206 "" 0 "" {TEXT 211 3 "If " }{TEXT 212 12 "p1, ..., pn " }{TEXT 211 17 " are elements of " }{TEXT 212 1 " R" }{TEXT 211 25 ", the differential ideal " }{TEXT 212 16 "I = [p1, . ., pn]" }{TEXT 211 18 " generated by the " }{TEXT 212 2 "pi" }{TEXT 211 35 " is the set of all the elements of " }{TEXT 212 1 "R" }{TEXT 211 56 " which are finite linear combinations (with elements of " } {TEXT 212 1 "R" }{TEXT 211 26 " for coefficients) of the " }{TEXT 212 2 "pi" }{TEXT 211 39 " and their derivatives up to any order." }} {PARA 203 "" 0 "" {TEXT 208 21 "A differential ideal " }{TEXT 213 1 "J " }{TEXT 208 4 " of " }{TEXT 213 1 "R" }{TEXT 208 37 " is said to be r adical if an element " }{TEXT 213 1 "r" }{TEXT 208 4 " of " }{TEXT 213 1 "R" }{TEXT 208 12 " belongs to " }{TEXT 213 1 "J" }{TEXT 208 56 " whenever one of its positive integral power belongs to " }{TEXT 213 1 "J" }{TEXT 208 1 "." }}{PARA 206 "" 0 "" {TEXT 211 3 "If " }{TEXT 212 12 "p1, ..., pn " }{TEXT 211 17 " are elements of " }{TEXT 212 1 " R" }{TEXT 211 33 ", the radical differential ideal " }{TEXT 212 17 "J \+ = \{p1, .., pn\} " }{TEXT 211 18 " generated by the " }{TEXT 212 2 "pi " }{TEXT 211 59 " is the smallest radical differential ideal containin g the " }{TEXT 212 2 "pi" }{TEXT 211 39 ". It is the set of all the el ements of " }{TEXT 212 1 "R" }{TEXT 211 30 ", a power of which belongs to " }{TEXT 212 16 "I = [p1, .., pn]" }{TEXT 211 1 "." }}{PARA 203 "" 0 "" {TEXT 208 83 "Another essential concept in differential algebra \+ is the notion of saturation. Let " }{TEXT 213 1 "S" }{TEXT 208 5 " and " }{TEXT 213 1 "H" }{TEXT 208 60 " be two sets of differential polyno mials. The saturation by " }{TEXT 213 1 "H" }{TEXT 208 48 " of the rad ical differential ideal generated by " }{TEXT 213 1 "S" }{TEXT 208 76 " contains all the differential polynomials that vanish for all the ze ros of " }{TEXT 213 1 "S" }{TEXT 208 38 " that are not zeros of any el ement of " }{TEXT 213 1 "H" }{TEXT 208 127 ". In simpler terms, this s aturation is the sharper differential ideal corresponding to the syste m of equations and inequations " }{TEXT 213 9 "S=0, H<>0" }{TEXT 208 1 "." }}{PARA 206 "" 0 "" {TEXT 211 4 "Let " }{TEXT 212 1 "S" }{TEXT 211 52 " be an non empty set of differential polynomials in " }{TEXT 212 1 "R" }{TEXT 211 5 " and " }{TEXT 212 1 "H" }{TEXT 211 25 " consis t of the elements " }{TEXT 212 11 "h1, ..., hm" }{TEXT 211 4 " of " } {TEXT 212 1 "R" }{TEXT 211 17 ". The saturation " }{TEXT 212 14 "\{S\} :H^infinity" }{TEXT 211 4 " of " }{TEXT 212 3 "\{S\}" }{TEXT 211 4 " b y " }{TEXT 212 1 "H" }{TEXT 211 44 " is the set of all differential po lynomials " }{TEXT 212 1 "r" }{TEXT 211 4 " of " }{TEXT 212 1 "R" } {TEXT 211 40 " for which there exists a power product " }{TEXT 212 1 " h" }{TEXT 211 8 " of the " }{TEXT 212 2 "hi" }{TEXT 211 13 "'s such th at " }{TEXT 212 3 "h*r" }{TEXT 211 12 " belongs to " }{TEXT 212 3 "\{S \}" }{TEXT 211 2 ". " }}{PARA 206 "" 0 "" {TEXT 212 14 "\{S\}:H^infini ty" }{TEXT 211 33 " is a radical differential ideal." }}{PARA 203 "" 0 "" {TEXT 208 21 "A differential ideal " }{TEXT 213 1 "P" }{TEXT 208 4 " of " }{TEXT 213 1 "R" }{TEXT 208 43 " is said to be prime if whene ver a product " }{TEXT 213 3 "p*q" }{TEXT 208 12 " belongs to " } {TEXT 213 1 "P" }{TEXT 208 9 ", either " }{TEXT 213 1 "p" }{TEXT 208 4 " or " }{TEXT 213 1 "q" }{TEXT 208 12 " belongs to " }{TEXT 213 1 "P " }{TEXT 208 1 "." }}{PARA 206 "" 0 "" {TEXT 211 41 "Given a parameter ised family of n-tuples " }{TEXT 212 11 "(v1,...,vn)" }{TEXT 211 66 " \+ of meromorphic functions, the set of differential polynomials in " } {TEXT 212 18 "R = F\{u1, ..., un\}" }{TEXT 211 58 " vanishing on this \+ tuple forms a prime differential ideal." }}{PARA 206 "" 0 "" {TEXT 211 29 "A radical differential ideal " }{TEXT 212 1 "J" }{TEXT 211 98 " is a finite intersection of prime differential ideals. This decompos ition is unique when minimal." }}}{SECT 0 {PARA 202 "" 0 "d_triangular " {TEXT 207 30 "Differentially triangular sets" }}{PARA 203 "" 0 "" {TEXT 208 34 "A set of differential polynomials " }{TEXT 213 11 "p1, . .., pn" }{TEXT 208 35 " in a polynomial differential ring " }{TEXT 213 1 "R" }{TEXT 208 67 " is said to be differentially triangular w.r. t. a given ranking if:" }}{PARA 206 "" 0 "" {TEXT 211 5 "- no " } {TEXT 212 2 "pi" }{TEXT 211 29 " belongs to the ground field." }} {PARA 206 "" 0 "" {TEXT 211 40 "- no proper derivative of the leader o f " }{TEXT 212 2 "pi" }{TEXT 211 12 " appears in " }{TEXT 212 2 "pj" } {TEXT 211 1 " " }{TEXT 212 15 "(i, j = 1 .. n)" }}{PARA 206 "" 0 "" {TEXT 211 21 "- the leaders of the " }{TEXT 212 2 "pi" }{TEXT 211 26 " 's are pairwise different." }}{PARA 203 "" 0 "" {TEXT 208 12 "The comm and " }{HYPERLNK 214 "differential_sprem" 2 "diffalg[differential_spre m]" "" }{TEXT 208 187 " implements a generalisation of the pseudo-divi sion algorithm to compute the reduction of a differential polynomial w .r.t. a differentially triangular systems of differential polynomials. " }}}{SECT 0 {PARA 202 "" 0 "regular_d_system" {TEXT 207 42 "Coherence and regular differential systems" }}{PARA 203 "" 0 "" {TEXT 208 32 "A differentially triangular set " }{TEXT 213 1 "R" }{TEXT 208 48 " is s aid to be coherent if all the well defined " }{HYPERLNK 214 "delta-pol ynomials" 2 "diffalg[delta_polynomial]" "" }{TEXT 208 142 " that can b e formed with any pair of its elements belong to the (non differential ) ideal determined by a limited differential prolongation of " }{TEXT 213 1 "A" }{TEXT 208 28 " (see precise definition in " }{HYPERLNK 214 "[Kolchin]" 2 "" "references" }{TEXT 208 2 ")." }}{PARA 206 "" 0 "" {TEXT 211 100 "The sufficient condition that appears commonly is that \+ all these delta-polynomials are reduced (via " }{HYPERLNK 214 "differe ntial_sprem" 2 "diffalg[differential_sprem]" "" }{TEXT 211 13 ") to ze ro by " }{TEXT 212 1 "A" }{TEXT 211 1 "." }}{PARA 203 "" 0 "" {TEXT 208 51 "A differential system of equations and inequations " }{TEXT 213 9 "A=0, H<>0" }{TEXT 208 7 ", with " }{TEXT 213 1 "A" }{TEXT 208 5 " and " }{TEXT 213 1 "H" }{TEXT 208 47 " finite subsets of different ial polynomials in " }{TEXT 213 1 "R" }{TEXT 208 27 ", is said to be r egular if " }{TEXT 213 1 "A" }{TEXT 208 49 " is a coherent differentia lly triangular set and " }{TEXT 213 1 "H" }{TEXT 208 4 " is " } {HYPERLNK 214 "partially reduced" 2 "diffalg[reduced]" "" }{TEXT 208 8 " w.r.t. " }{TEXT 213 1 "A" }{TEXT 208 5 " and " }{TEXT 213 1 "H" } {TEXT 208 43 " contains the separants of the elements of " }{TEXT 213 1 "A" }{TEXT 208 1 "." }}{PARA 203 "" 0 "" {TEXT 208 30 "A regular dif ferential system " }{TEXT 213 9 "A=0, H<>0" }{TEXT 208 40 " defines th e regular differential ideal " }{TEXT 213 14 "[A]:H^infinity" }{TEXT 208 1 "." }}{PARA 206 "" 0 "" {TEXT 211 95 "Testing triviality or memb ership to a regular differential ideal are purely algebraic problems." }}{PARA 206 "" 0 "" {TEXT 211 167 "Every regular differential ideal i s radical and is an intersection of prime differential ideals which ha ve the same parametric set (arbitrary constants and functions)." }}} {SECT 0 {PARA 202 "" 0 "d_characteristic_set" {TEXT 207 72 "Differenti al characteristic sets and characterisable differential ideals" }} {PARA 203 "" 0 "" {TEXT 208 2 "A " }{HYPERLNK 214 "differentially tria ngular set" 2 "" "d_triangular" }{TEXT 208 1 " " }{TEXT 213 1 "C" } {TEXT 208 67 " is a differential characteristic set when we have the e quivalence:" }}{PARA 206 "" 0 "" {TEXT 211 26 "a differential polynomi al " }{TEXT 212 1 "p" }{TEXT 211 9 " belongs " }{TEXT 212 14 "[C]:H^in finity" }{TEXT 211 16 " if and only if " }{HYPERLNK 214 "differential_ sprem" 2 "diffalg[differential_sprem]" "" }{TEXT 211 1 " " }{TEXT 212 8 "(p, C)=0" }{TEXT 211 1 "." }}{PARA 206 "" 0 "" {TEXT 211 6 "where " }{TEXT 212 1 "H" }{TEXT 211 61 " is the set of the initials and separ ants of the elements of " }{TEXT 212 1 "C" }{TEXT 211 1 "." }}{PARA 203 "" 0 "" {TEXT 208 21 "A characteristic set " }{TEXT 213 1 "C" } {TEXT 208 48 " defines the characterisable differential ideal " } {TEXT 213 14 "[C]:H^infinity" }{TEXT 208 2 ", " }{TEXT 213 1 "H" } {TEXT 208 58 " the set of the initials and separants of the elements o f " }{TEXT 213 1 "C" }{TEXT 208 1 "." }}{PARA 206 "" 0 "" {TEXT 211 211 "Characterisable differential ideals are regular differential idea ls. They are radical and are the intersection of prime differential id eals which have the same parametric set (arbitrary constants and funct ions)." }}{PARA 203 "" 0 "" {TEXT 208 60 "The non singular zeros of a \+ differential characteristic set " }{TEXT 213 1 "C" }{TEXT 208 39 " are the zeros for which no element of " }{TEXT 213 1 "H" }{TEXT 208 8 " v anish." }}{PARA 206 "" 0 "" {TEXT 211 84 "The non singular zeros of a \+ characterisable differential ideal can be expanded into " }{HYPERLNK 214 "formal integral power series" 2 "diffalg[power_series_solution]" "" }{TEXT 211 52 " up to any order. Convergence is not secured though. " }}{PARA 203 "" 0 "" {TEXT 208 313 "A prime differential ideal is a c haracterisable differential ideal for any ranking. A characterisable d ifferential ideal is prime if its defining characteristic set is irred ucible. A sufficient condition for that is that all the differential p olynomials in the characteristic set have degree one in their leaders. " }}}{SECT 0 {PARA 202 "" 0 "characteristic_decomposition" {TEXT 207 28 "Characteristic decomposition" }}{PARA 203 "" 0 "" {TEXT 208 73 "An y radical differential ideal can be decomposed into an intersection of " }{HYPERLNK 214 "characterisable differential ideals" 2 "" "d_charac teristic_set" }{TEXT 208 185 ". The characterisable differential ideal s coming into the decomposition are called (characterisable) component s and the representation obtained is called a characteristic decomposi tion." }}{PARA 206 "" 0 "" {TEXT 211 61 "This representation is neithe r unique nor need to be minimal." }}{PARA 206 "" 0 "" {TEXT 211 63 "A \+ characteristic decomposition of a radical differential ideal " }{TEXT 212 1 "J" }{TEXT 211 15 " allows one to " }{HYPERLNK 214 "test members hip" 2 "diffalg[belongs_to]" "" }{TEXT 211 4 " to " }{TEXT 212 1 "J" } {TEXT 211 1 "." }}{PARA 203 "" 0 "" {TEXT 208 12 "The command " } {HYPERLNK 214 "Rosenfeld_Groebner" 2 "diffalg[Rosenfeld_Groebner]" "" }{TEXT 208 197 " computes a characteristic decomposition of a radical \+ differential ideal generated by a finite set of differential polynomia ls. It can also compute the characteristic decomposition of a saturati on." }}{PARA 203 "" 0 "" {TEXT 208 88 "The algorithm proceeds in two f undamental steps. It first computes a decomposition into " }{HYPERLNK 214 "regular differential ideals" 2 "" "regular_d_ideals" }{TEXT 208 37 ". The algorithm used is described in " }{HYPERLNK 214 "[Boulier et al. 1997]" 2 "" "references" }{TEXT 208 143 ". The second step consis ts in computing characteristic decompositions of these regular differe ntial ideals. The algorithm used is described in " }{HYPERLNK 214 "[Hu bert 2000]" 2 "" "references" }{TEXT 208 1 "." }}{PARA 203 "" 0 "" {TEXT 208 165 "When considering the radical differential ideal generat ed by a unique differential polynomial one can obtain a minimal charac teristic decomposition with the command " }{HYPERLNK 214 "essential_co mponents" 2 "diffalg[essential_components]" "" }{TEXT 208 24 ". The al gorithm used by " }{TEXT 213 20 "essential_components" }{TEXT 208 17 " is described in " }{HYPERLNK 214 "[Hubert 1999]" 2 "" "references" } {TEXT 208 1 "." }}}{SECT 0 {PARA 202 "" 0 "ideals_representation" {TEXT 207 65 "Representation of characterisable and radical differenti al ideals" }}{PARA 203 "" 0 "" {TEXT 208 83 "A characterisable differe ntial ideal is represented by a MAPLE table, appearing as " }{TEXT 213 15 "characterisable" }{TEXT 208 63 ", that records the defining ch aracteristic set, as well as the " }{HYPERLNK 214 "differential polyno mial ring" 2 "diffalg[differential_ring]" "" }{TEXT 208 39 " with resp ect to which it was computed." }}{PARA 203 "" 0 "" {TEXT 208 191 "A ra dical differential ideal is representable by a characteristic decompos ition, that is an intersection of characterisable differential ideals. This intersection is represented by a list of " }{TEXT 213 15 "charac terisable" }{TEXT 208 8 " tables." }}{PARA 203 "" 0 "" {TEXT 208 74 "C haracterisable and radical differential ideals are built by the functi on " }{HYPERLNK 214 "Rosenfeld-Groebner" 2 "diffalg[Rosenfeld_Groebner ]" "" }{TEXT 208 1 "." }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 14 "with(diffalg):" }{MPLTEXT 1 209 58 "\nR := differential_ring(ranking= [[z,y]], derivations=[x]);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>%\"RG% )ODE_ringG" }{TEXT 210 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 55 "S := [-y[]+x* y[x]+y[x]^2+z[x], -z[]+x*z[x]+y[x]*z[x]];" }} {PARA 205 "" 1 "" {XPPMATH 20 "6#>%\"SG7$,*&%\"yG6\"!\"\"*&%\"xG\"\"\" &F(6#F,F-F-*$F.\"\"#F-&%\"zGF/F-,(&F3F)F**&F,F-F2F-F-*&F.F-F2F-F-" } {TEXT 210 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 44 "Radic al_d_ideal := Rosenfeld_Groebner(S, R);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>%0Radical_d_idealG7%%0characterisableGF&F&" }{TEXT 210 1 " " }} }{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 46 "Characterisable_d_ideal := Radical_d_ideal[1];" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>%8Charact erisable_d_idealG%0characterisableG" }{TEXT 210 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 33 "indices(Characterisable_d_ideal);" }} {PARA 205 "" 1 "" {XPPMATH 20 "6(7#%/Set_of_leadersG7#%*EquationsG7#%/ RepresentationG7#%2Differential_ringG7#%%TypeG7#%,InequationsG" } {TEXT 210 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 30 "Chara cterisable_d_ideal[Type];" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#%3Differ ential_IdealG" }{TEXT 210 1 " " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 54 "evalb(Characterisable_d_ideal[Differential_ring] = \+ R);" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#%%trueG" }{TEXT 210 1 " " }}} {PARA 206 "" 0 "" {TEXT 211 21 "The other entries of " }{TEXT 212 23 " Characterisable_d_ideal" }{TEXT 211 40 " are in internal notations. Th e entries " }{TEXT 212 9 "Equations" }{TEXT 211 5 " and " }{TEXT 212 11 "Inequations" }{TEXT 211 31 " are readable via the commands " } {HYPERLNK 214 "equations and inequations" 2 "diffalg[equations]" "" } {TEXT 211 4 " or " }{HYPERLNK 214 "rewrite_rules" 2 "diffalg[equations ]" "" }{TEXT 211 2 " ." }}{PARA 203 "" 0 "" {TEXT 208 28 "If the envir onment variable " }{TEXT 213 17 "_Env_diffalg_char" }{TEXT 208 18 " is set to false, " }{TEXT 213 18 "Rosenfeld_Groebner" }{TEXT 208 91 " re presents the radical differential ideals as intersection of regular di fferential ideals." }}{PARA 206 "" 0 "" {TEXT 211 121 "Regular differe ntial ideal are tables with an identical structure to characterisable \+ differential ideals. They appear as " }{TEXT 212 7 "regular" }{TEXT 211 1 "." }}}{SECT 0 {PARA 202 "" 0 "references" {TEXT 207 10 "Referen ces" }}{PARA 203 "" 0 "" {TEXT 208 48 "The reference books of differen tial algebra are:" }}{PARA 206 "" 0 "" {TEXT 211 68 "- Differential Al gebra by J.F. Ritt, Dover Publications Inc. (1966)." }}{PARA 206 "" 0 "" {TEXT 211 74 "- An Introduction to Differential Algebra by I. Kapla nsky, Hermann (1970)." }}{PARA 206 "" 0 "" {TEXT 211 91 "- Differentia l Algebra and Algebraic Groups by E. Kolchin, Academic Press, New York (1973)." }}{PARA 203 "" 0 "" {TEXT 208 60 "The algorithms implemented in this package are presented in:" }}{PARA 206 "" 0 "" {TEXT 211 179 "- Representation for the radical of a finitely generated differential ideal by F. Boulier, D. Lazard, F. Ollivier and M. Petitot in the pro ceedings of ISSAC95, pp. 158-166 (1995)." }}{PARA 206 "" 0 "" {TEXT 211 166 "- Representation for the radical of a finitely generated diff erential ideal, by F. Boulier, D. Lazard, F. Ollivier and M. Petitot, \+ Research Report LIFL IT-306 (1997)." }}{PARA 206 "" 0 "" {TEXT 211 148 "- Essential Components of an Algebraic Differential Equation, by \+ E. Hubert, Journal of Symbolic Computation, vol.28 number 4-5 pages 65 7-680 (1999)." }}{PARA 206 "" 0 "" {TEXT 211 157 "- Factorisation free decomposition algorithms in differential algebra, by E. Hubert, Journ al of Symbolic Computation, vol.29 number 4-5 pages 641-662 (2000)." } }{PARA 206 "" 0 "" {TEXT 211 143 "- Differential Algebra for Derivatio ns with Nontrivial Commutation rules, by E. Hubert, Journal of Pure an d Applied Algebra, to appear ( 2005)." }}}{SECT 0 {PARA 202 "" 0 "seea lso" {TEXT 207 10 "See Also: " }}{PARA 204 "" 0 "" {HYPERLNK 214 "diff alg" 2 "diffalg" "" }{TEXT 219 2 ". " }}}{PARA 209 "" 0 "" }} {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }