Multivariate Graded Lagrange Interpolation

The constitution theory of a properly posed set of nodes forthe multivariate polynomial graded interpolation is studied deeply inthe paper. On the basis of Lagrange interpolation which along the al-gebraic curve without multiple factors,we give the approach of gradedLagrange interpolation which along the algebraic curve without mul-tiple factors. Futhermore,using this result we give a basic method toconstruct the graded Lagrange interpolation in R~2. In addition,usingweak Grobner basis' method which is a new mathematic concept andthe theory for constructing the properly posed set of nodes for inter-polation on plane algebraic curve,we give the method to construct theproperly posed set of nodes for graded interpolation on plane algebraiccurve accordingly. At the same time, using the results of algebraic vari-ety and ideal in algebraic geometry, we study the geometrical structureof properly posed set of nodes for graded interpolation on algebraic hy-persurface, more over,we give a Hyperplane Superposition Process toconstruct the properly posed set of nodes for graded interpolation onalgebraic hypersurface,therefore we make clear the geometrical struc-ture of properly posed set of nodes for multivariate graded interpola-tion basically.On the other hand,when we study the effect of movinginterpolation nodes to the property for interpolation,we improve andprove an important theorem gao, thus we acquire the general conclu-sion.