| Let n be a positive integer and m a nonnegative integer. A ranking {dollar}le{dollar} is a total order of {dollar}{lcub}rm I!N{rcub}sp{lcub}m{rcub}times{lcub}1,...,n{rcub}{dollar} with {dollar}(alpha,i)le(beta,j)Longrightarrow(alpha+gamma,i)leq (beta+gamma,j){dollar} for {dollar}alpha,beta,gammain{lcub}rm I!N{rcub}sp{lcub}m{rcub}{dollar} and {dollar}i,jin{lcub}1,...,n{rcub}.{dollar} The ranking {dollar}le{dollar} is positive if {dollar}alpha,i)ge(0,i){dollar} for all {dollar}(alpha,i)in{lcub}rm I!N{rcub}sp{lcub}m{rcub}times{lcub}1,...,n{rcub}.{dollar} For {dollar}n=1, ((alphasb1,...,alphasb{lcub}m{rcub}),1)in{lcub}rm I!N{rcub}sp{lcub}m{rcub}times{lcub}1{rcub}{dollar} corresponds to the monomial {dollar}xsbsp{lcub}1{rcub}{lcub}alphasb1{rcub}{lcub}cdots{rcub}xsbsp{lcub}m{rcub}{lcub}alphasb{lcub}m{rcub}{rcub}.{dollar} Positive rankings in this case, known as monomial orderings, are a critical component of many algorithms in polynomial algebra, notably Buchberger's algorithm. For {dollar}nge1, ((alphasb1,...,alphasb{lcub}m{rcub}),i)in{lcub}rm I!N{rcub}sp{lcub}m{rcub}times{lcub}1,...,n{rcub}{dollar} corresponds to the derivative {dollar}partialsp{lcub}alphasb1+...+alphasb{lcub}m{rcub}{rcub}usp{lcub}i{rcub}/partial xsbsp{lcub}1{rcub}{lcub}alphasb1{rcub}cdotspartial xsbsp{lcub}m{rcub}{lcub}alphasb{lcub}m{rcub}{rcub}.{dollar} Such rankings are key to elimination algorithms for systems of partial differential equations, such as Janet's algorithm and differential Grobner basis methods, and to Buchberger's algorithm for free modules over polynomial rings.; We describe how to specify an arbitrary ranking by concrete data. In so doing we introduce previously unknown rankings. A large class of rankings is key to making differential and algebraic elimination algorithms more efficient. Furthermore, by the Riquier Existence Theorem, our new rankings provide new Cauchy-type Existence and Uniqueness Theorems for formal and local analytic solutions to analytic systems of partial differential equations.; The Riquier-Janet theory parameterizes the local solutions at a point to a system of real or complex analytic linear partial differential equations (and some nonlinear systems), except where an explicit analytic equation vanishes. We give a new development of the Riquier-Janet theory motivated by algebraic Grobner basis theory. We show that the output of a modified version of the Janet algorithm is independent of conventional choices except the ranking. We generalize the Riquier Existence Theorem for formal solutions to arbitrary positive rankings.; The reduced involutive form algorithm of Reid, Wittkopf and Boulton applies to more general nonlinear systems. We give an algebraic version of this geometric algorithm and prove an explicit Existence and Uniqueness Theorem for our version. For polynomially nonlinear systems, this Existence and Uniqueness Theorem also applies to the output of the algorithm of Reid et al. |
