The Study Of The Multivariate Rational Interpolation Problems On Algebraic Mainfolds

Interpolation is one of the important methods of function approximation. Interpolation by rational functions sometimes is better than by polynomials. While there are many difficulties when dealing with it by traditional method because of the complicated structure of rational functions.Because the rational interpolation problems on the Algebraic Manifolds has many important applications in many aspects of cryptography, this paper uses the view,theory and method of constructive algebraic geometry, to deal with the rational interpolation on the algebraic manifolds. Two issues were discussed: Cauchy-type multivariate rational interpolation on the algebraic manifolds, multivariate osculatory rational interpolation on the algebraic manifolds.For the Cauchy-type multivariate rational interpolation problem on the algebraic manifolds, we discussed the interpolation problems on an irreducible algebraic mainfold S. we request that notes satisfy:Definition 1. Let V = {X₀,...,X_m+n} C S be interpolation nodes, in which S is an irreducible algebraic mainfold.N(I(S)) is the corresponding quotient ring's standards-bases on(?). t₀,...,t_m+n are the first m+n+1 monomials from low to high on (?).For any given polynomial g in the I(â…¤)\I(S), if LT(g) (?)t_M, in which M=max{m, n}. we said thatfor the (m, n) type Rational interpolation problem, V is on " General Position" w.r.t. S.For the interpolation notes satisfied Definition l,we haveTheorem 2. Let V={X₀,...,X_m+n}(?)5 satisfy Definition 1. G_s is the reduced Gr(o|¨)bner bases of I(S). Give the interpolation datas f₀,..., f_m+n∈R, on V. If the interpolation function meeting the conditions exists,then the interpolation basis composing the numerator and the denominator of R(X) is unique, and they can be just costituted by the M lowest which can not be divised by LT(G_s).We also discussed the interpolation problem on a range of irreducible algebraic mainfolds∪S_i.Theorem 3. Giving some algebraic manifolds 5 and its minimal decomposition S=S₁∪S₂∪...∪S_t, Interpolation nodes are V={X₀,...,X_m+n} on the " General position" w.r.t. 5, and different each other. Give the interpolation datas f₀,..., f_m+n∈R, on V. If the interpolation function meeting the conditionsR(X_i)=f_i, i=0,...,m+n.exists,then the interpolation basis composing the numerator and the denominator of R(X) is unique, And they can be just costituted by the M lowest mononials in the N(I(∪S_i)).When the requirements of rational interpolation function is the (m, n) type,then because we only need the first M+1 mononials of the minimal interpolation basis, of which M=max{m, n}, we have improved the conditions for the termination of Algorithm 1.According to these theories, for the Cauchy-type rational interpolation problems on Algebraic Manifolds ,through improving the well-known MMM algorithm in [13], this paper gave a much faster algorithm-the algorithm 2.For the osculatory rational interpolation problems on the algebraic manifolds,because the multivariate osculatory rational interpolation is analysised basing on the Hermite interpolation, and the Hermite interpolation is not only related to the location of the interpolation notes, That is, the shape of algebraic manifolds, but also depends on the interpolation conditions of these nodes. At this point, we have given a method to calculate the minimal interpolation bases:For the (m, n) type rational interpolation problem on an irreducible alge- braic mainfold S, we defined the power factor s,make (?)f∈G(S),have f^s∈I_int,So (LT(f))^s,(?)f∈G(S) can be skipped in the MMM algorithm in order to simplify the algorithm 1.As for the interpolation problem on reducible algebraic manifold,we put it into a number of irreducible algebraic manifolds S_i, Then in accordance with the above approach,we can also skip through directly LT(F) to simplify the algorithm 1,in whichAs the same as the Cauchy-type rational interpolation problems, the requirements of rational interpolation function is the (m, n) type,then because we only need the first M+1 mononials of the minimal interpolation basis, of which M=max{m,n}, we have improved the conditions for the termination of Algorithm 1.According to these theories, algebraic manifold for the osculatory rational interpolation problems, this paper gave a much faster algorithm-the algorithm 3.On the other hand, in the last two sections, we have discussed the existence of these two issues, as well as the expression. The problem od rational interpolation is essentially nonlinear. For the case of multivariate osculatory rational interpolation we prove it is equivalent to a linear problem,which machs the problem much easier.Further we prove that for the given interpolation conditions if we choose the minimal interpolation bases as the supports of the denominator and the numerator of the rational interpolation functions,the rational interpolation functions exist for almost all given interpolation data,which is to say, the probability that the rational interpolation functions exist is equal to 1. So basically in theory, we completely resolved the problem of the Cauchy-type multivariate rational interpolation, multivariate osculatory rational interpolation on the algebraic manifolds ,such as the existence, the unique and the constructure of interpolation function,.Finally, in each chapter, we give some examples of these theories and algorithms for the analysis, we will not put them here.