bibkey: nigamphillips2007pyramids authors: Nilima Nigam and Joel Phillips year: 2007 title: Higher-order finite elements on pyramids doi: 10.48550/arXiv.math/0610206 url: https://arxiv.org/abs/math/0610206v3 claim: “The reference pyramid has nonnegative coordinates with x,y at most 1-z; equations (2.3) and (6.1) give a projective map and first-order rational shape functions.” strata_touched:

• D5/S1/Words/AdmissibleWords/PathStableSetPolytope license: citation-only triage: anchor

Reference pyramid and the scope of rational shape functions

The primary text consumed here is the fixed author version arXiv:math/0610206v3, with the title above. Equation (1.2), PDF p. 2, defines

$$\Omega =\{(x,y,z):x,y,z\ge 0,x\le 1-z,y\le 1-z\}.$$

Theorem 1.1, PDF p. 3, excludes purely polynomial high-order $H^1(\Omega )$-conforming finite elements satisfying the compatibility property specified in the paper. The proof uses a rational function with polynomial face traces and derives a contradiction from a proposed polynomial representation. This is a statement about that finite-element model and compatibility requirement, not a claim that all functions on a pyramid must be rational or that arbitrary vertex data determine them.

The determinant sequel uses this reference domain and the explicit projective shape-function bridge. Equation (2.3), PDF p. 3, maps infinite-pyramid coordinates $(u,v,h)$ to $(u/(1+h),v/(1+h),h/(1+h))$. Section 6.1, equation (6.1), PDF p. 26, prints the five pullbacks

$$({\tilde{\pi }}_1,{\tilde{\pi }}_2,{\tilde{\pi }}_3,{\tilde{\pi }}_4,{\tilde{\pi }}_5)=\frac{((u-1)(v-1),u(v-1),(u-1)v,uv,h)}{1+h}.$$

The volume keeps these printed signs explicit: its nonnegative weights in order $(0,2,5,25,3)$ are $({\tilde{\pi }}_1,-{\tilde{\pi }}_2,-{\tilde{\pi }}_3,{\tilde{\pi }}_4,{\tilde{\pi }}_5)$. The elementary substitution $r=1/(1+h)$ supplies that correspondence. The paper attributes its first-order input to Gradinaru and Hiptmair; this note does not assign discovery priority to either this volume or the higher-order construction.

The continuous apex extension, coordinate Lebesgue integral coefficients $1/16$ and $1/12$, and interior bubble calculation have their own elementary proofs in the volume. Those exact coefficients are not attributed to this source. No Appendix Table 2, journal-version title equivalence, approximation estimate or native FIB response theorem is consumed from this v3 text.