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Boundaries of algebraic elements such as polycons are defined by irreducible polynomials with real coefficients. We cannot restrict the analysis to polynomials over the reals because this field is not algebraically closed. Points of intersection of curves of polynomials with only real coefficients are in general points with complex coordinates which may reduce in specific cases to real coordinates. These intersection points are of vital concern in this development, and for this reason we perform our analysis with the complex coefficient field.

A more complicated situation is shown in Fig. 11 in which node 1 is a side node of element m and a vertex node for elements p and q. 2 Fig. 11. H y b r i d nodes on a c u r v e d side. 16 RATIO NAL FINITE ELEMENT BASIS Points 4 and 5 are side nodes for p and q,respectively, but these points are not nodes of element m. The restricted values at these points are expressed in terms of the wedges for element m: 3 - i=1 i=l 3 CI We will be concerned primarily with patchwork approximation over collections of well-set polycons.

This point is called a "side node". Thus a polycon with r conic and s linear sides has 2r+s nodes. This is equal to the order of the polycon. We will develop the theory for construction of wedge functions for these nodes. It is preferable to partibion the domain of interest into elements such that no node is a vertex of one element and a side node of another element. In certain situations, such "hybrid" nodes are introduced. For example, referring to Fig. 10, where the element size is reduced along line (2;3), we observe that node 1 is hybrid.

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