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Source. H.-J. Rack and R. Vajda, Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant, Stud. Univ. Babeş-Bolyai Math. 60 (2015), no. 2, 151--171; the edition read is named on the source card.

Statement

Setting. $b>1$ is the right endpoint of the parameter interval $[1,b]$ in Theorem 4.2; $t=0.4177913013\ldots$ is the constant (3.5) of the optimal canonical system $-1<-t<t<1$. In the proof of Theorem 5.2 (p. 165) $b$ is the point $x>1$ where the Lebesgue function of that canonical system, $x(1-t+t^2-x^2)/((-1+t)t)$ there (6.6), equals $\Lambda_4^*$.

Lemma 4.3 (p. 157). $b$ is the unique positive root of the degree-18 integer polynomial

$$\begin{aligned} P_{18}(x)={}&-121+220x-1014x^2+1344x^3+3283x^4-5166x^5+4502x^6\\ &+15692x^7-84178x^8+7868x^9+210676x^{10}-25694x^{11}-310732x^{12}\\ &+34154x^{13}+255377x^{14}-8450x^{15}-124700x^{16}+26875x^{18}, \end{aligned}\qquad(4.2)$$

and numerically $b=1.0433133411\ldots$ (4.3); (4.6) on p. 158 gives more digits.

Lemma 4.4 (pp. 157-158). $b$ is given by radicals in terms of $t$ as

$$b=b(t)=\bigl(p-\sqrt{D}\bigr)^{1/3}+\bigl(p+\sqrt{D}\bigr)^{1/3},\qquad p=\frac{t+t^3}{2+2t},\quad D=-\frac1{27}\bigl(1+(-1+t)t\bigr)^3+\frac{(t+t^3)^2}{4(1+t)^2},\qquad(4.4)$$

which is the printed (4.4) with its two repeated subexpressions named $p$ and $D$. Substituting (3.5) for $t$ gives a long expression for $b$ by radicals alone, printed as (4.5) on p. 158.

Proof pointer

Lemma 4.3: Section 6.4, p. 166. Mathematica's RootReduce applied to the equation (6.6) $=\Lambda_4^*$, with $\Lambda_4^*$ and $t$ given as roots of (3.1) and of $Q_3^*(x^2)$ (input (6.10)), returns $P_{18}$ (output (6.11)). Lemma 4.4: Section 6.5, p. 167. With $\Lambda_4^*=(1+t^2)/(1-t^2)$ (3.9), the equation becomes the cubic $(t+t^3)+(1+t^3)x+(-1-t)x^3=0$ (6.13), solved by Cardan's formula.

Read depth

Claims checked: both lemmas and (4.2) were read on the page image of the print, and the coefficients of (4.2) were matched against the printed output (6.11). The computer-algebra steps were not re-run, and (4.5) was not transcribed or checked. Nothing here is independently reviewed.

Dependencies

The cubic constants (3.1)-(3.9), recorded by the paper from its references [23], [24], [29], [30]; see the Theorem 5.2 page.

Bears on

• Problem 1129: the lemmas identify the constant $b$ that bounds the parameter ranges in the paper's description of all minimizing four-node systems (Theorem 5.2).