Exceptional polynomials

This page gives exceptional polynomials according to the definition in my paper Distance to the discriminant. They are all homogeneous polynomials in three variables and therefore their zero locus are algebraic curves in the projective plane. These polynomials maximize the distance to the discriminant for the Bombieri norm, among polynomials of the same norm. As a consequence (see the paper), they can all be written as sums of powers of linear forms in the directions which correspond to critical points of P on the unit sphere with the minimum absolute critical value.

This page presents two kinds of polynomials: exact ones and numerical approximations. The former are normalized to be at distance one to the discriminant, while the latter are normalized to have Bombieri norm 1. In both cases, we give two expressions:

The original form $P$ arising from the construction of the curve.
The writing $P_{1}$ as a sum of power of linear forms.
For exact polynomials we have $P = P_{1}$ . For numerical polynomials, $\frac{∥ P - P_{1} ∥}{∥ P ∥}$ gives us an interesting error which tells us if the polynomial is optimized enough. For instance, if it is less than $\frac{dist (P, Δ)}{∥ P ∥}$ , we know that both polynomials have zero locus which are ambient-isotopic. When this is not the case, the optimization process to maximize the distance to the discriminant is not finished yet and the polynomial is marked with a ✖ sign.

We provide images of the zero locus curves that can be zoomed and dragged. The curve is projected onto a disc shown in light grey using the stereographic projection $(x, y, z) \mapsto (\frac{x}{1 + z}, \frac{y}{1 + z})$ of the hemisphere $z > 0$ . We also highlight with a cross the critical points with the least critical absolute value on the unit sphere. These are the directions of the linear forms in the expression as a sum of linear forms to the power d.

Degree	Topology	Nb. of linear forms	$\frac{dist (P, Δ)}{∥ P ∥}$	$\frac{∥ P - P_{1} ∥}{∥ P ∥}$	Good

Exact polynomials

Degree 2, ⟨1⟩ ✔

\begin{array}{rcl} P & = & x^{2} + y^{2} - z^{2} \\ = & x^{2} + y^{2} - z^{2} \\ ‖ P ‖ & = & \sqrt{3} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \end{array}

Degree 3, ⟨J ∐ 1⟩ ✔

\begin{array}{rcl} P & = & z (- 3 x^{2} - 3 y^{2} + z^{2}) \\ = & \frac{5 z^{3}}{3} - \frac{{(- \sqrt{3} x + z)}^{3}}{6} - \frac{{(\sqrt{3} x + z)}^{3}}{6} - \frac{{(- \sqrt{3} y + z)}^{3}}{6} - \frac{{(\sqrt{3} y + z)}^{3}}{6} \\ ‖ P ‖ & = & \sqrt{7} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \end{array}

Degree 4, ⟨4⟩ ✔

\begin{array}{rcl} P & = & (- x + y + z) (x - y + z) (x + y - z) (6 x + 6 y + 6 z) - {(x^{2} + y^{2} + z^{2})}^{2} \\ = & - \frac{13 {(x - y)}^{4}}{3} - \frac{13 {(x + y)}^{4}}{3} - \frac{13 {(x - z)}^{4}}{3} - \frac{13 {(x + z)}^{4}}{3} - \frac{13 {(y - z)}^{4}}{3} - \frac{13 {(y + z)}^{4}}{3} + \frac{31 {(x - y - z)}^{4}}{12} + \frac{31 {(x - y + z)}^{4}}{12} + \frac{31 {(x + y - z)}^{4}}{12} + \frac{31 {(x + y + z)}^{4}}{12} \\ ‖ P ‖ & = & \sqrt{197} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \end{array}

Degree 4, ⟨3⟩ ✔

\begin{array}{rcl} P & = & (- x + y + z) (x - y + z) (x + y - z) (2 x + 2 y + 2 z) + {(x^{2} + y^{2} + z^{2})}^{2} \\ = & - 3 x^{4} - 3 y^{4} - 3 z^{4} + \frac{{(x - y)}^{4}}{2} + \frac{{(x + y)}^{4}}{2} + \frac{{(x - z)}^{4}}{2} + \frac{{(x + z)}^{4}}{2} + \frac{{(y - z)}^{4}}{2} + \frac{{(y + z)}^{4}}{2} \\ ‖ P ‖ & = & \sqrt{21} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \end{array}

Degree 6, ⟨10⟩ V1 ✔

\begin{array}{rcl} P & = & - {(x^{2} + y^{2} + z^{2})}^{3} + \frac{2 (- x (φ + 2) + z (\sqrt{5} + 3 + \frac{1}{φ})) (x (φ + 2) + z (\sqrt{5} + 3 + \frac{1}{φ})) (x (\sqrt{5} + 3 + \frac{1}{φ}) - y (φ + 2)) (x (\sqrt{5} + 3 + \frac{1}{φ}) + y (φ + 2)) (y (\sqrt{5} + 3 + \frac{1}{φ}) - z (φ + 2)) (y (\sqrt{5} + 3 + \frac{1}{φ}) + z (φ + 2))}{{(- \frac{\sqrt{3} (φ + 2)}{3} + \frac{\sqrt{3} (\sqrt{5} + 3 + \frac{1}{φ})}{3})}^{3} {(\frac{\sqrt{3} (φ + 2)}{3} + \frac{\sqrt{3} (\sqrt{5} + 3 + \frac{1}{φ})}{3})}^{3}} \\ = & \frac{127 {(- φ x + \frac{z}{φ})}^{6}}{30} + \frac{127 {(φ x + \frac{z}{φ})}^{6}}{30} + \frac{127 {(- φ y + \frac{x}{φ})}^{6}}{30} + \frac{127 {(φ y + \frac{x}{φ})}^{6}}{30} + \frac{127 {(- φ z + \frac{y}{φ})}^{6}}{30} + \frac{127 {(φ z + \frac{y}{φ})}^{6}}{30} + \frac{127 {(x - y - z)}^{6}}{30} + \frac{127 {(x - y + z)}^{6}}{30} + \frac{127 {(x + y - z)}^{6}}{30} + \frac{127 {(x + y + z)}^{6}}{30} - (\frac{219}{20} - \frac{73 \sqrt{5}}{15}) (x^{6} {(1 + \sqrt{5})}^{6} + y^{6} {(1 + \sqrt{5})}^{6} + z^{6} {(1 + \sqrt{5})}^{6} + {(- φ x - y (φ + 1) + z)}^{6} + {(- φ x + y (φ + 1) + z)}^{6} + {(φ x - y (φ + 1) + z)}^{6} + {(φ x + y (φ + 1) + z)}^{6} + {(- φ y + x - z (φ + 1))}^{6} + {(- φ y + x + z (φ + 1))}^{6} + {(φ y + x - z (φ + 1))}^{6} + {(φ y + x + z (φ + 1))}^{6} + {(- φ z - x (φ + 1) + y)}^{6} + {(- φ z + x (φ + 1) + y)}^{6} + {(φ z - x (φ + 1) + y)}^{6} + {(φ z + x (φ + 1) + y)}^{6}) \\ ‖ P ‖ & = & \sqrt{2311} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \end{array}

Degree 6, ⟨10⟩ V2 ✔

\begin{array}{rcl} P & = & 1458 (- \frac{4 x^{2}}{9} - \frac{4 y^{2}}{9} + \frac{5 z^{2}}{9}) (- \frac{4 x^{2}}{9} + \frac{5 y^{2}}{9} - \frac{4 z^{2}}{9}) (\frac{5 x^{2}}{9} - \frac{4 y^{2}}{9} - \frac{4 z^{2}}{9}) + {(x^{2} + y^{2} + z^{2})}^{3} \\ = & - \frac{9236 {(\frac{\sqrt{2} x}{2} - \frac{\sqrt{2} y}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} x}{2} + \frac{\sqrt{2} y}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} x}{2} - \frac{\sqrt{2} z}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} x}{2} + \frac{\sqrt{2} z}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} y}{2} - \frac{\sqrt{2} z}{2})}^{6}}{3} - \frac{9236 {(\frac{\sqrt{2} y}{2} + \frac{\sqrt{2} z}{2})}^{6}}{3} + \frac{15255 {(\frac{x}{3} - \frac{2 y}{3} - \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{x}{3} - \frac{2 y}{3} + \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{x}{3} + \frac{2 y}{3} - \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{x}{3} + \frac{2 y}{3} + \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} - \frac{2 y}{3} - \frac{z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} - \frac{2 y}{3} + \frac{z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} - \frac{y}{3} - \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} - \frac{y}{3} + \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} + \frac{y}{3} - \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} + \frac{y}{3} + \frac{2 z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} + \frac{2 y}{3} - \frac{z}{3})}^{6}}{4} + \frac{15255 {(\frac{2 x}{3} + \frac{2 y}{3} + \frac{z}{3})}^{6}}{4} - 6744 {(\frac{\sqrt{3} x}{3} - \frac{\sqrt{3} y}{3} - \frac{\sqrt{3} z}{3})}^{6} - 6744 {(\frac{\sqrt{3} x}{3} - \frac{\sqrt{3} y}{3} + \frac{\sqrt{3} z}{3})}^{6} - 6744 {(\frac{\sqrt{3} x}{3} + \frac{\sqrt{3} y}{3} - \frac{\sqrt{3} z}{3})}^{6} - 6744 {(\frac{\sqrt{3} x}{3} + \frac{\sqrt{3} y}{3} + \frac{\sqrt{3} z}{3})}^{6} \\ ‖ P ‖ & = & \sqrt{91213} \\ ‖ P - P_{1} ‖ & = & 0 \\ dist (P, Δ) & = & 1 \end{array}

Approximated polynomials

Degree 5, ⟨𝐽 ∐ 6⟩ ✔

\begin{array}{rcl} P & = & - 0.00106873415699291 x^{5} + 0.0298190870015163 x^{4} y + 0.0298190870015163 x^{4} z - 0.0965192797742038 x^{3} y^{2} + 0.531316220915451 x^{3} y z - 0.0965192797742038 x^{3} z^{2} - 0.0965192797742038 x^{2} y^{3} - 3.08382139247772 x^{2} y^{2} z - 3.08382139247772 x^{2} y z^{2} - 0.0965192797742038 x^{2} z^{3} + 0.0298190870015163 x y^{4} + 0.531316220915451 x y^{3} z - 3.08382139247772 x y^{2} z^{2} + 0.531316220915451 x y z^{3} + 0.0298190870015163 x z^{4} - 0.00106873415699291 y^{5} + 0.0298190870015163 y^{4} z - 0.0965192797742038 y^{3} z^{2} - 0.0965192797742038 y^{2} z^{3} + 0.0298190870015163 y z^{4} - 0.00106873415699291 z^{5} \\ ≃ & - 41.4816247609779 {(- x - 0.0740590551607519 y - 0.0740590551607519 z)}^{5} + 19.0067688827384 {(- 0.403906402377617 x - y + 0.0601498422398717 z)}^{5} + 19.0067688827384 {(- 0.403906402377617 x + 0.0601498422398717 y - z)}^{5} - 19.0067688827384 {(- 0.0601498422398717 x + y + 0.403906402377617 z)}^{5} + 11.2705241276892 {(0.0407005400461146 x - y + 0.0407005400461146 z)}^{5} + 11.2705241276892 {(0.0407005400461146 x + 0.0407005400461146 y - z)}^{5} + 19.0067688827384 {(0.0601498422398717 x - 0.403906402377617 y - z)}^{5} + 41.4816247609779 {(0.0740590551607519 x + 0.0740590551607519 y + z)}^{5} + 41.4816247609779 {(0.0740590551607519 x + y + 0.0740590551607519 z)}^{5} - 4.10506254927065 {(0.151988408912245 x - y - z)}^{5} + 4.10506254927065 {(x - 0.151988408912245 y + z)}^{5} - 19.0067688827384 {(x - 0.0601498422398717 y + 0.403906402377617 z)}^{5} - 11.2705241276892 {(x - 0.0407005400461146 y - 0.0407005400461146 z)}^{5} - 19.0067688827384 {(x + 0.403906402377617 y - 0.0601498422398717 z)}^{5} + 4.10506254927065 {(x + y - 0.151988408912245 z)}^{5} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.44491908908016 \cdot 10^{-14} \\ dist (P, Δ) & = & 0.00249150281902411 \end{array}

Degree 5, ⟨𝐽 ∐ 5⟩ ✔

\begin{array}{rcl} P & = & 0.138282576385147 x^{5} + 1.49131316544416 \cdot 10^{-99} x^{4} y + 0.587838979473938 x^{4} z - 1.38282576385147 x^{3} y^{2} - 2.44910491937703 \cdot 10^{-99} x^{3} y z - 2.4721962043497 \cdot 10^{-16} x^{3} z^{2} + 9.94197259244896 \cdot 10^{-100} x^{2} y^{3} + 1.17567795894788 x^{2} y^{2} z - 9.16959832802434 \cdot 10^{-100} x^{2} y z^{2} - 1.09348571906218 x^{2} z^{3} + 0.691412881925735 x y^{4} - 2.4491125754873 \cdot 10^{-99} x y^{3} z - 2.42766577840164 \cdot 10^{-16} x y^{2} z^{2} + 2.2249450310098 \cdot 10^{-99} x y z^{3} + 2.09232162391621 \cdot 10^{-16} x z^{4} - 4.97105866270899 \cdot 10^{-100} y^{5} + 0.587838979473938 y^{4} z + 3.05659310627729 \cdot 10^{-100} y^{3} z^{2} - 1.09348571906218 y^{2} z^{3} + 1.45896158593391 \cdot 10^{-105} y z^{4} + 0.520200768722881 z^{5} \\ ≃ & 0.854248544976042 {(- x - 6.36138300480644 \cdot 10^{-100} y + 0.432857562680349 z)}^{5} - 0.296055048062032 {(- x + 0.726542528005361 y - 0.535041372047788 z)}^{5} - 0.714545660551176 {(- 0.910622886844367 x - 1.69246128588907 \cdot 10^{-100} y - z)}^{5} + 0.623486541071559 {(- 0.860611265166907 x - 0.625270684224257 y - z)}^{5} + 0.623486541071559 {(- 0.860611265166907 x + 0.625270684224257 y - z)}^{5} + 0.66468499593733 {(- 0.324919696232906 x - y + 0.455133375634235 z)}^{5} - 0.660854148606108 {(- 0.324919696232906 x - y + 0.98842629974994 z)}^{5} - 0.714545660551176 {(- 0.281397947501684 x - 0.86605383042084 y - z)}^{5} - 0.714545660551176 {(- 0.281397947501684 x + 0.86605383042084 y - z)}^{5} + 0.660854148606108 {(0.324919696232906 x - y - 0.98842629974994 z)}^{5} - 0.66468499593733 {(0.324919696232906 x - y - 0.455133375634235 z)}^{5} - 0.714545660551176 {(0.736709390923868 x - 0.535250703287117 y - z)}^{5} - 0.714545660551176 {(0.736709390923868 x + 0.535250703287117 y - z)}^{5} + 0.84932516656563 {(x + 3.28309933270347 \cdot 10^{-100} y - 0.940049273254687 z)}^{5} + 0.296055048062032 {(x + 0.726542528005361 y + 0.535041372047788 z)}^{5} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 4.0699497755238 \cdot 10^{-12} \\ dist (P, Δ) & = & 0.0230355306974538 \end{array}

Degree 6, ⟨9 ∐ 1⟨1⟩⟩ ✔

\begin{array}{rcl} P & = & 9.61142335107702 \cdot 10^{-5} x^{6} + 3.94443008510763 \cdot 10^{-10} x^{5} y + 2.67267233662849 \cdot 10^{-10} x^{5} z + 1.8911537585164 x^{4} y^{2} + 2.58331622515706 x^{4} y z + 0.875510022522386 x^{4} z^{2} + 2.6699939491284 \cdot 10^{-10} x^{3} y^{3} - 6.20517332816258 \cdot 10^{-10} x^{3} y^{2} z - 2.05949268852434 \cdot 10^{-9} x^{3} y z^{2} - 9.31641981872975 \cdot 10^{-10} x^{3} z^{3} - 1.26028860154198 x^{2} y^{4} + 1.7222108157218 x^{2} y^{3} z + 1.75102004537569 x^{2} y^{2} z^{2} - 1.04016485661513 x^{2} y z^{3} - 0.64644514778324 x^{2} z^{4} - 1.32826799300697 \cdot 10^{-10} x y^{5} + 6.55961777154313 \cdot 10^{-10} x y^{4} z - 3.68042052089827 \cdot 10^{-10} x y^{3} z^{2} - 9.32752465571435 \cdot 10^{-10} x y^{2} z^{3} + 3.18455509935993 \cdot 10^{-10} x y z^{4} + 1.98003078321575 \cdot 10^{-10} x z^{5} + 0.21019227166814 y^{6} - 0.861105408365832 y^{5} z + 0.875510021681551 y^{4} z^{2} + 0.346721619732491 y^{3} z^{3} - 0.646445147562631 y^{2} z^{4} - 1.37166228772424 \cdot 10^{-10} y z^{5} + 9.61142335107702 \cdot 10^{-5} z^{6} \\ ≃ & 364.983457349339 {(- 1.0 x - 2.66900897282968 \cdot 10^{-13} y + 1.53079345072811 \cdot 10^{-10} z)}^{6} + 29.8388637293865 {(- x - 0.577350269340767 y + 0.775541601120748 z)}^{6} + 211.938095216215 {(- x - 0.375545562266798 y + 0.504461887612111 z)}^{6} - 161.483452605925 {(- x + 0.577350269201071 y - 0.668049808814068 z)}^{6} + 29.8388637719511 {(- x + 0.57735026920297 y - 0.775541600630207 z)}^{6} - 67.9275376335915 {(- 0.844215781451024 x + y - 0.332934338839026 z)}^{6} + 153.977396069108 {(- 0.577350269189982 x + y + 1.94424953367616 \cdot 10^{-10} z)}^{6} + 153.977396069393 {(- 0.577350269189271 x - y - 1.76641511972671 \cdot 10^{-11} z)}^{6} - 236.500422647515 {(- 0.360582465505215 x - y - 0.270433765988846 z)}^{6} - 236.500422647242 {(- 0.360582465423023 x + y + 0.270433766099294 z)}^{6} - 382.775590986068 {(- 8.82968642354794 \cdot 10^{-11} x - y - 0.578548105812708 z)}^{6} + 70.729158489878 {(1.02547115751821 \cdot 10^{-10} x + y + 0.671638728256721 z)}^{6} + 3.1833702276807 {(1.53079345074169 \cdot 10^{-10} x - 1.06045404174552 \cdot 10^{-10} y + 1.0 z)}^{6} + 290.237619920579 {(0.16584579980329 x - y - 0.478708252551352 z)}^{6} + 290.237619920733 {(0.165845799949302 x + y + 0.478708252500534 z)}^{6} - 67.9275376337752 {(0.844215781348179 x + y - 0.33293433909734 z)}^{6} + 66.9368667695977 {(x - 0.821893267639955 y + 0.611297052185078 z)}^{6} + 211.938095412868 {(x - 0.375545562208188 y + 0.50446188722794 z)}^{6} - 310.320670890538 {(x - 0.179416457913439 y + 0.258462645343354 z)}^{6} - 310.320670743024 {(x + 0.179416457928187 y - 0.258462645669989 z)}^{6} - 161.483452407458 {(x + 0.577350269319868 y - 0.668049809257068 z)}^{6} + 66.9368666942565 {(x + 0.82189326779467 y - 0.611297052605911 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 5.42369272468914 \cdot 10^{-13} \\ dist (P, Δ) & = & 9.61142335107125e-5 \end{array}

Degree 6, ⟨6 ∐ 1⟨2⟩⟩ ✔

\begin{array}{rcl} P & = & 0.00589370417177452 x^{6} - 0.00810722420831899 x^{5} y - 0.10447541397529 x^{5} z - 0.13246170003432 x^{4} y^{2} + 0.507503425686876 x^{4} y z + 0.48496566263209 x^{4} z^{2} - 0.340529737547038 x^{3} y^{3} + 1.06649896955336 x^{3} y^{2} z - 4.25866713675959 x^{3} y z^{2} - 0.102094774223009 x^{3} z^{3} - 0.387024873194602 x^{2} y^{4} + 0.82973870683771 x^{2} y^{3} z + 6.87333566012994 x^{2} y^{2} z^{2} + 0.500807048068058 x^{2} y z^{3} - 0.021632585240089 x^{2} z^{4} - 0.198554452044143 x y^{5} + 1.15205593386011 x y^{4} z + 1.04043383730238 x y^{3} z^{2} + 0.961863155460903 x y^{2} z^{3} + 0.157899579839491 x y z^{4} + 0.000589006101698444 x z^{5} - 0.0260996888277111 y^{6} + 0.200758307624054 y^{5} z - 0.0915693420332 y^{4} z^{2} + 0.169052103282536 y^{3} z^{3} + 0.0328301791661001 y^{2} z^{4} + 0.000389369942468445 y z^{5} - 0.000247824170496405 z^{6} \\ ≃ & 87.6664686456415 {(- x - 0.150729359722728 y - 0.115783979001648 z)}^{6} + 265.695755990963 {(- x + 0.065650241518947 y - 0.0859866326669049 z)}^{6} - 153.79999337981 {(- x + 0.327903635249339 y - 0.0335761031721888 z)}^{6} + 83.6672235426775 {(- 0.195405846721067 x + y + 0.119139788183464 z)}^{6} + 127.022023420699 {(- 0.195116491394136 x + y + 0.403494638876342 z)}^{6} - 107.213151262721 {(- 0.147291769569149 x - 0.025716993584979 y + z)}^{6} + 406.791376401183 {(- 0.0941546536962571 x - 0.136470699034113 y + z)}^{6} - 121.237821904953 {(0.000831354098122493 x + 0.00406978931983558 y - z)}^{6} - 106.983611766237 {(0.0280259173527562 x - 0.149336631492256 y + z)}^{6} + 22.3654640248944 {(0.129250780640282 x - 0.671894154008624 y - z)}^{6} - 28.1542350628732 {(0.175126643994255 x - y - 0.796355326718402 z)}^{6} - 124.693820298526 {(0.214945809118715 x + 0.307841181349247 y - z)}^{6} - 301.312716030954 {(0.251479143205027 x - y - 0.182387164170782 z)}^{6} + 22.0029855816638 {(0.329948439608101 x + 0.470999288294642 y - z)}^{6} + 160.649748053158 {(0.432550554274225 x - y - 0.0966045811728143 z)}^{6} + 22.145418796939 {(0.6800810289777 x + 0.102391327909679 y + z)}^{6} - 50.8327784091383 {(0.775571712917709 x - y - 0.0444100591692092 z)}^{6} + 56.5191181177349 {(x - 0.704853915275409 y - 0.00203725283830958 z)}^{6} - 353.199224403675 {(x + 0.0963892033420925 y + 0.175084637386817 z)}^{6} + 133.630759068277 {(x + 0.150357435173335 y + 0.39766673301071 z)}^{6} - 28.6371826321118 {(x + 0.169226117718793 y + 0.791708077838355 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.62071918034527 \cdot 10^{-13} \\ dist (P, Δ) & = & 0.000248848440022236 \end{array}

Degree 6, ⟨2 ∐ 1⟨6⟩⟩ ✔

\begin{array}{rcl} P & = & - 0.396663870421844 x^{6} + 0.21640136821547 x^{5} y - 0.0109419425863422 x^{5} z - 0.817302526479224 x^{4} y^{2} + 0.48072714803358 x^{4} y z + 1.27206642925114 x^{4} z^{2} + 0.0508999223953116 x^{3} y^{3} - 0.175349776021804 x^{3} y^{2} z - 0.451149966359061 x^{3} y z^{2} + 0.0232569761630139 x^{3} z^{3} - 0.163029808841459 x^{2} y^{4} + 1.04638338228826 x^{2} y^{3} z + 1.51195475956509 x^{2} y^{2} z^{2} - 1.04470250583701 x^{2} y z^{3} - 1.35783369451186 x^{2} z^{4} - 0.234906246896496 x y^{5} - 0.166627696213582 x y^{4} z + 0.0221136877114013 x y^{3} z^{2} + 0.181835573825987 x y^{2} z^{3} + 0.235225289182257 x y z^{4} - 0.0123739559359566 x z^{5} + 0.194998705706446 y^{6} + 0.566460762192207 y^{5} z - 0.00753890289944145 y^{4} z^{2} - 1.13007440315571 y^{3} z^{3} - 0.670730783712748 y^{2} z^{4} + 0.566435465154493 y z^{5} + 0.482446648620434 z^{6} \\ ≃ & 5899.18615919742 {(- x - 0.153713895669976 y - 0.931201436252078 z)}^{6} + 6426.08488080274 {(- x - 0.113795140982786 y - 0.950918189688898 z)}^{6} - 2784.69877035772 {(- x - 0.00198844342967822 y + 0.983369335070547 z)}^{6} + 11386.0057152613 {(- x + 0.0685421657295613 y + 0.96837319059512 z)}^{6} + 6651.72783599151 {(- x + 0.117662727225871 y + 0.94850868789565 z)}^{6} + 15282.449101315 {(- 0.95487687220295 x + 0.372309894295322 y + z)}^{6} - 12283.7780535856 {(- 0.773860992874597 x + 0.624620329967185 y + z)}^{6} - 1178.71157913803 {(- 0.424977514643304 x + y - 0.921732598097279 z)}^{6} + 8228.00135863891 {(- 0.300637679903004 x - y - 0.980601084315638 z)}^{6} - 8637.51375922199 {(- 0.107499146706803 x + y + 0.998557107581799 z)}^{6} + 10250.5989662291 {(0.482681602053948 x - 0.854600981819339 y - z)}^{6} + 3450.56919088966 {(0.506922361950585 x - 0.977234802278101 y + z)}^{6} - 9242.36245168653 {(0.5827266931497 x - 0.836811739222528 y + z)}^{6} - 5161.02207862197 {(0.658524996768161 x - 0.682249366831095 y + z)}^{6} + 17902.626733883 {(0.669123124275664 x - 0.699694420920674 y + z)}^{6} - 7005.34187682588 {(0.688983720559287 x + 0.881556040275767 y + z)}^{6} - 7176.85403354259 {(0.771066115394857 x - 0.573282702674997 y + z)}^{6} + 8332.79310064146 {(0.952461619185363 x + 0.608855047150633 y + z)}^{6} - 24264.0846894138 {(x - 0.169966209444015 y - 0.970604507798814 z)}^{6} - 17235.781047003 {(x + 0.285184179247319 y + 0.946652598889968 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 7.45538634751606 \cdot 10^{-11} \\ dist (P, Δ) & = & 6.56498373940561e-7 \end{array}

Degree 6, ⟨1 ∐ 1⟨9⟩⟩ ✖

\begin{array}{rcl} P & = & - 0.463676008413115 x^{6} + 1.26390330377364 \cdot 10^{-8} x^{5} y - 4.63257086820353 \cdot 10^{-7} x^{5} z - 0.650659383368547 x^{4} y^{2} + 0.262615746075259 x^{4} y z + 1.46209055277665 x^{4} z^{2} - 8.65065554707416 \cdot 10^{-9} x^{3} y^{3} - 4.63703003390089 \cdot 10^{-7} x^{3} y^{2} z + 1.38303903248024 \cdot 10^{-7} x^{3} y z^{2} + 9.72574226759599 \cdot 10^{-7} x^{3} z^{3} - 0.117427822291915 x^{2} y^{4} + 0.704813684540312 x^{2} y^{3} z + 1.66715542428863 x^{2} y^{2} z^{2} - 0.445952509119675 x^{2} y z^{3} - 1.52545873396788 x^{2} z^{4} - 1.60444576368352 \cdot 10^{-8} x y^{5} - 7.21090619761633 \cdot 10^{-8} x y^{4} z + 2.24607818138239 \cdot 10^{-7} x y^{3} z^{2} + 5.80365292711415 \cdot 10^{-7} x y^{2} z^{3} - 1.2530094535893 \cdot 10^{-7} x y z^{4} - 5.06804485221378 \cdot 10^{-7} x z^{5} + 0.0855830564465105 y^{6} + 0.405460169459421 y^{5} z + 0.407757382820572 y^{4} z^{2} - 0.600021803746839 y^{3} z^{3} - 0.994311051098061 y^{2} z^{4} + 0.186036994408672 y z^{5} + 0.527208683439629 z^{6} \\ ≃ & 1850555.33536835 {(- 0.987851014886197 x - 0.106139848855809 y + z)}^{6} + 1850557.13964424 {(- 0.987850684956379 x + 0.106139877291987 y - z)}^{6} - 2374262.60458365 {(- 0.980532959724258 x - 0.260438506340935 y + z)}^{6} + 574945.415617187 {(- 0.977062851629122 x - 0.3014385919025 y + z)}^{6} - 1494722.60067477 {(- 0.956483807554877 x + 0.132187366982547 y + z)}^{6} - 251102.556612314 {(- 0.954196775835736 x - 0.368561929641538 y + z)}^{6} - 251102.793094531 {(- 0.954196444521271 x + 0.368561915918526 y - z)}^{6} + 1250247.19326426 {(- 0.848183366353608 x - 0.411727964293163 y - z)}^{6} - 1080726.14789132 {(- 0.643485624820289 x - 0.675413990297642 y - z)}^{6} + 976119.653309522 {(- 0.349228101904382 x - 0.865220308774889 y - z)}^{6} - 940224.535399759 {(- 6.06365882716484 \cdot 10^{-8} x + 0.934658609081793 y + z)}^{6} - 5257.98474577754 {(6.86626322520366 \cdot 10^{-8} x + y - 0.553685855372674 z)}^{6} + 5465.53453059173 {(6.90898645192625 \cdot 10^{-8} x + y - 0.558880262824073 z)}^{6} + 976119.316858747 {(0.349228246450881 x - 0.86522037462946 y - z)}^{6} - 1080725.46151275 {(0.643485826196302 x - 0.675414091549598 y - z)}^{6} + 1250246.14663035 {(0.848183630151682 x - 0.41172806096371 y - z)}^{6} - 1494724.01174178 {(0.956483498679039 x + 0.132187301951209 y + z)}^{6} + 1019052.94414165 {(0.967090225033273 x - 0.327324104804175 y + z)}^{6} + 1019051.97145515 {(0.967090558515698 x + 0.327324112152367 y - z)}^{6} + 574945.970062263 {(0.977062516154575 x - 0.301438588638958 y + z)}^{6} - 2374264.9023209 {(0.980532625028345 x - 0.260438509678899 y + z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.02741897789258 \cdot 10^{-8} \\ dist (P, Δ) & = & 5.82146436611808e-9 \end{array}

Degree 6, ⟨5 ∐ 1⟨5⟩⟩ ✖

\begin{array}{rcl} P & = & - 0.00173148646203087 y^{6} - 0.117098678607153 y^{5} (0.707106781186548 x - 0.707106781186548 z) + 0.00125289713693927 y^{5} (0.707106781186548 x + 0.707106781186548 z) + 0.0789527163346608 y^{4} (0.707106781186548 x - 0.707106781186548 z) (0.707106781186548 x + 0.707106781186548 z) - 0.000900144612987245 y^{4} {(x - z)}^{2} + 9.52332712069707 \cdot 10^{-5} y^{4} {(x + z)}^{2} + 0.021497373461506 y^{3} (0.707106781186548 x - 0.707106781186548 z) {(x + z)}^{2} - 0.192050007893459 y^{3} (0.707106781186548 x + 0.707106781186548 z) {(x - z)}^{2} + 0.000624364191753918 y^{3} {(x - z)}^{3} + 6.31464516107628 \cdot 10^{-5} y^{3} {(x + z)}^{3} + 0.00898855461939147 y^{2} (0.707106781186548 x - 0.707106781186548 z) {(x + z)}^{3} - 0.000603250888414344 y^{2} (0.707106781186548 x + 0.707106781186548 z) {(x - z)}^{3} - 9.20431138588168 \cdot 10^{-6} y^{2} {(x - z)}^{4} + 0.507483427966932 y^{2} {(x - z)}^{2} {(x + z)}^{2} + 2.70647584982688 \cdot 10^{-6} y^{2} {(x + z)}^{4} + 0.000336429843237311 y (0.707106781186548 x - 0.707106781186548 z) {(x + z)}^{4} + 0.000253394533300613 y (0.707106781186548 x + 0.707106781186548 z) {(x - z)}^{4} - 2.37526695040164 \cdot 10^{-7} y {(x - z)}^{5} - 0.0265350636551808 y {(x - z)}^{3} {(x + z)}^{2} + 0.170967500801209 y {(x - z)}^{2} {(x + z)}^{3} + 4.71767832987528 \cdot 10^{-9} y {(x + z)}^{5} + 3.34042462883763 \cdot 10^{-7} (0.707106781186548 x - 0.707106781186548 z) {(x + z)}^{5} + 1.96104713401188 \cdot 10^{-5} (0.707106781186548 x + 0.707106781186548 z) {(x - z)}^{5} - 1.30198011887763 \cdot 10^{-8} {(x - z)}^{6} - 0.00482794020306371 {(x - z)}^{4} {(x + z)}^{2} + 0.539976473641131 {(x - z)}^{3} {(x + z)}^{3} + 0.00518917319685263 {(x - z)}^{2} {(x + z)}^{4} + 6.29794845460381 \cdot 10^{-12} {(x + z)}^{6} \\ ≃ & 21841555.543418 {(- 0.991083014475548 x - 0.427275813539191 y - z)}^{6} - 8673533.00849556 {(- 0.988750352368422 x - 0.536992582449404 y - z)}^{6} + 1988953.01134833 {(- 0.987212671244603 x - 0.612314443229877 y - z)}^{6} - 70186382.9934559 {(- 0.979658439513886 x - 0.236017289935104 y + z)}^{6} + 39143455.6861762 {(- 0.959520220788682 x - 0.321514587382398 y + z)}^{6} - 17300852.3897331 {(- 0.937919947122169 x - 0.396281822928437 y + z)}^{6} + 4306604.20836193 {(- 0.921008596496029 x - 0.447434768837249 y + z)}^{6} - 43295845.2554917 {(0.993784105712659 x + 0.300410045815559 y + z)}^{6} + 111150212.500513 {(0.994267552007015 x + 0.153876146630478 y - z)}^{6} + 73728621.0578512 {(0.996403877872837 x + 0.172888387316721 y + z)}^{6} - 112698081.56301 {(0.998696588447908 x + 0.0590939591593067 y + z)}^{6} - 23847980.7229699 {(x - 0.126566395308295 y + 0.99471939169477 z)}^{6} + 96356784.0566063 {(x - 0.109721328344797 y + 0.996268150375628 z)}^{6} + 54448086.0493098 {(x - 0.0980265769591207 y + 0.997929600873293 z)}^{6} - 220732478.863822 {(x - 0.0844365698579566 y + 0.997840949533234 z)}^{6} + 160713574.029161 {(x - 0.0292827928764943 y + 0.999367443614603 z)}^{6} + 25862498.377619 {(x + 0.0110133872867473 y - 0.994808478588007 z)}^{6} - 104150506.611908 {(x + 0.0246971271010069 y - 0.994257797222141 z)}^{6} - 58921956.3105163 {(x + 0.0345116827291138 y - 0.993369720725046 z)}^{6} + 236136515.336229 {(x + 0.0447368676679939 y - 0.994414407783434 z)}^{6} - 165662877.85036 {(x + 0.0872317532335946 y - 0.99759123590325 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 0.125267921379197 \\ dist (P, Δ) & = & 7.33183227836020e-11 \end{array}

Degree 7, ⟨𝐽 ∐ 15⟩ ✔

\begin{array}{rcl} P & = & - 1.74415402873375 \cdot 10^{-10} x^{7} + 0.193985061996143 x^{6} y - 0.0674698195570803 x^{6} z + 1.47457241517766 \cdot 10^{-8} x^{5} y^{2} - 3.34288587749177 \cdot 10^{-8} x^{5} y z + 1.01491178382161 \cdot 10^{-8} x^{5} z^{2} + 3.19653872230365 x^{4} y^{3} - 0.53710506334235 x^{4} y^{2} z - 1.1261213376319 x^{4} y z^{2} + 0.322178912499216 x^{4} z^{3} - 6.16682520046227 \cdot 10^{-8} x^{3} y^{4} - 2.28770197209417 \cdot 10^{-7} x^{3} y^{3} z + 1.98897825631941 \cdot 10^{-8} x^{3} y^{2} z^{2} + 1.10575615971318 \cdot 10^{-7} x^{3} y z^{3} - 3.04774765965267 \cdot 10^{-8} x^{3} z^{4} - 0.412696946813008 x^{2} y^{5} - 2.87922246276088 x^{2} y^{4} z - 4.6943444130989 x^{2} y^{3} z^{2} + 0.797444140566674 x^{2} y^{2} z^{3} + 1.81877730525078 x^{2} y z^{4} - 0.487300026164741 x^{2} z^{5} + 3.89755112948739 \cdot 10^{-9} x y^{6} + 4.20794560515217 \cdot 10^{-8} x y^{5} z + 1.50063760088953 \cdot 10^{-7} x y^{4} z^{2} + 1.73813500346237 \cdot 10^{-7} x y^{3} z^{3} - 3.22693680867766 \cdot 10^{-8} x y^{2} z^{4} - 7.91808693360586 \cdot 10^{-8} x y z^{5} + 2.17225891352041 \cdot 10^{-8} x z^{6} + 0.0132679769537078 y^{7} + 0.187064674905035 y^{6} z + 0.978383145672103 y^{5} z^{2} + 2.24737285428801 y^{4} z^{3} + 1.84740166165716 y^{3} z^{4} - 0.468810634173273 y^{2} z^{5} - 0.810329908503586 y z^{6} + 0.222620135268511 z^{7} \\ ≃ & 13.5338736804469 {(- x - 0.0325440203289802 y + 0.702663067722303 z)}^{7} - 737.583529677243 {(- x + 0.743039351238574 y + 0.878303244044841 z)}^{7} + 2379.42068576429 {(- 0.973785960204335 x - 0.379977873746931 y - z)}^{7} + 661.484304216664 {(- 0.878865467022381 x + y + 0.682048508367528 z)}^{7} + 2712.37399704992 {(- 0.81056160296501 x + 0.347844590607477 y + z)}^{7} + 2776.33928247353 {(- 0.571963646054319 x - y - 0.350841169052581 z)}^{7} - 2776.3393345973 {(- 0.571963633287145 x + y + 0.35084117878 z)}^{7} - 4023.11923601096 {(- 0.558694059828708 x + 0.344795883897711 y + z)}^{7} + 15.7168985315213 {(- 3.78709129855932 \cdot 10^{-9} x + y - 0.65747483679439 z)}^{7} - 25906.5374311872 {(1.01336532328173 \cdot 10^{-10} x + y - 0.240534781256537 z)}^{7} - 5486.17016256206 {(2.00869668275224 \cdot 10^{-10} x - y + 0.272939103914304 z)}^{7} - 6816.42620861485 {(1.0144968087762 \cdot 10^{-8} x + 0.349255322569399 y + z)}^{7} - 17593.4136984705 {(0.0754484577637087 x - y + 0.190956316165553 z)}^{7} + 17593.4136548997 {(0.075448458917823 x + y - 0.190956317640392 z)}^{7} + 12224.3120087946 {(0.19289823684724 x - y + 0.0670226309931826 z)}^{7} - 12224.3119313934 {(0.192898240460782 x + y - 0.0670226346517863 z)}^{7} - 5877.09335304249 {(0.27890524824487 x - 0.347843056651376 y - z)}^{7} + 5877.09313902575 {(0.278905269979105 x + 0.347843057153089 y + z)}^{7} - 7031.52383557802 {(0.35516074255618 x - y - 0.112982110160897 z)}^{7} + 7031.5237536053 {(0.355160749944215 x + y + 0.112982103724524 z)}^{7} - 4023.11894253981 {(0.558694085919819 x + 0.344795884870971 y + z)}^{7} + 2712.37370999562 {(0.810561635503026 x + 0.34784459206559 y + z)}^{7} + 661.484285134095 {(0.878865488055219 x + y + 0.68204849478557 z)}^{7} + 2379.42098829064 {(0.973785922083203 x - 0.379977871411549 y - z)}^{7} - 1392.08351380718 {(x - 0.496890579079 y - 0.950997197283233 z)}^{7} - 522.166671535481 {(x - 0.345201983525361 y - 0.992400347566472 z)}^{7} + 13.5338749076353 {(x - 0.032544024596597 y + 0.70266303996806 z)}^{7} + 522.166745110989 {(x + 0.345201971887549 y + 0.992400308938053 z)}^{7} + 1392.0837093636 {(x + 0.496890564418115 y + 0.950997159546204 z)}^{7} - 737.583632249909 {(x + 0.74303933178777 y + 0.878303207943774 z)}^{7} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 4.51722369306773 \cdot 10^{-11} \\ dist (P, Δ) & = & 2.38287955322755e-6 \end{array}

Degree 8, ⟨18 ∐ 1⟨3⟩⟩ ✔

\begin{array}{rcl} P & = & 3.42639808625191 \cdot 10^{-8} x^{8} - 5.20218194718691 \cdot 10^{-9} x^{7} y - 3.34927900961968 \cdot 10^{-9} x^{7} z + 2.14927084494929 x^{6} y^{2} + 2.76750793316963 x^{6} y z + 0.890891344032204 x^{6} z^{2} - 8.7090761580403 \cdot 10^{-9} x^{5} y^{3} + 1.99506657847714 \cdot 10^{-8} x^{5} y^{2} z + 5.118016639116 \cdot 10^{-8} x^{5} y z^{2} + 2.23976367917429 \cdot 10^{-8} x^{5} z^{3} + 0.716423769876259 x^{4} y^{4} + 4.6125132170289 x^{4} y^{3} z + 1.14924961321223 x^{4} y^{2} z^{2} - 3.52509693886025 x^{4} y z^{3} - 1.63800689164346 x^{4} z^{4} - 1.75990922440709 \cdot 10^{-9} x^{3} y^{5} - 8.52627334429744 \cdot 10^{-9} x^{3} y^{4} z + 4.80459271614805 \cdot 10^{-8} x^{3} y^{3} z^{2} + 3.33511890999359 \cdot 10^{-8} x^{3} y^{2} z^{3} - 3.73752474256241 \cdot 10^{-8} x^{3} y z^{4} - 2.39056725244219 \cdot 10^{-8} x^{3} z^{5} - 1.194039148444 x^{2} y^{6} + 0.922502623527613 x^{2} y^{5} z + 3.68829033047973 x^{2} y^{4} z^{2} - 2.35006459612284 x^{2} y^{3} z^{3} - 3.27601379108108 x^{2} y^{2} z^{4} + 1.14917690065212 x^{2} y z^{5} + 1.02387845537451 x^{2} z^{6} + 1.74698495308753 \cdot 10^{-9} x y^{7} - 7.23401247549844 \cdot 10^{-9} x y^{6} z - 3.17088271888009 \cdot 10^{-9} x y^{5} z^{2} + 2.62547268472772 \cdot 10^{-8} x y^{4} z^{3} - 5.24450473680407 \cdot 10^{-9} x y^{3} z^{4} - 2.39160381153039 \cdot 10^{-8} x y^{2} z^{5} + 4.32834637838548 \cdot 10^{-9} x y z^{6} + 7.1086594668879 \cdot 10^{-9} x z^{7} + 0.23880789236505 y^{8} - 0.922502642618129 y^{7} z + 0.721621944432765 y^{6} z^{2} + 1.17503233171612 y^{5} z^{3} - 1.63800687629417 y^{4} z^{4} - 0.383058989845222 y^{3} z^{5} + 1.02387845225685 y^{2} z^{6} + 5.1202972337329 \cdot 10^{-9} y z^{7} - 0.215868035709408 z^{8} \\ ≃ & 19767.4637029659 {(- x - 0.577350269428545 y + 0.898662175380796 z)}^{8} - 153180.377996312 {(- x - 0.40222910268907 y + 0.627475853538732 z)}^{8} - 426329.319728964 {(- x - 0.132670269054444 y + 0.206173247306412 z)}^{8} - 426329.317078407 {(- x + 0.132670269162058 y - 0.206173251233114 z)}^{8} + 272236.890761712 {(- x + 0.268667658116027 y - 0.418188881987154 z)}^{8} + 19767.4631672846 {(- x + 0.577350271388767 y - 0.898662182191393 z)}^{8} - 41092.7521843926 {(- x + 0.783778164276347 y - 0.739650846526161 z)}^{8} + 573.820851296882 {(- 0.912476857041548 x - 0.526818759308686 y + z)}^{8} + 573.820867073842 {(- 0.912476850136677 x + 0.526818757502219 y - z)}^{8} - 71304.9522647089 {(- 0.768917474779285 x + y - 0.257816378855582 z)}^{8} + 273020.24625107 {(- 0.267230986469105 x + y + 0.418038708156371 z)}^{8} + 254650.19728118 {(- 0.0511317895435347 x + y + 0.709338972579217 z)}^{8} + 254650.197280711 {(- 0.0511317868763487 x - y - 0.709338972771967 z)}^{8} + 871.577734232877 {(- 1.78511270603537 \cdot 10^{-9} x + y + 0.949093013537398 z)}^{8} + 62474.9456376641 {(- 1.46340159910457 \cdot 10^{-9} x + y + 0.77826427910536 z)}^{8} - 327499.460329716 {(1.42048095419891 \cdot 10^{-9} x - y - 0.755473398429139 z)}^{8} - 20.0966855998858 {(1.88494360867706 \cdot 10^{-9} x + 0.756514554911236 y - z)}^{8} - 257616.796570734 {(0.142117608372451 x + y + 0.587997718609434 z)}^{8} - 257616.796572055 {(0.142117610582529 x - y - 0.587997718073774 z)}^{8} + 273020.246248437 {(0.267230984899405 x + y + 0.418038709163393 z)}^{8} - 243453.638673955 {(0.413042123529946 x + y + 0.221130392225297 z)}^{8} - 243453.638677583 {(0.413042124357549 x - y - 0.221130390669172 z)}^{8} + 158964.060122797 {(0.57735026918531 x - y - 2.70951196481765 \cdot 10^{-10} z)}^{8} + 158964.060119486 {(0.577350269191323 x + y + 2.44552674210029 \cdot 10^{-9} z)}^{8} - 20.0966860457291 {(0.655160818011842 x + 0.378257274370473 y + z)}^{8} - 20.0966856489968 {(0.655160823393317 x - 0.378257275306833 y - z)}^{8} - 71304.9522627307 {(0.76891747575752 x + y - 0.257816375960368 z)}^{8} + 22606.2513912136 {(x - 0.9986604666285 y + 0.570771505982319 z)}^{8} - 103622.875606472 {(x - 0.57735027132453 y + 0.872345539971407 z)}^{8} + 101688.205465191 {(x - 0.511129454998596 y + 0.795587590928071 z)}^{8} + 272236.89419475 {(x + 0.268667657688013 y - 0.418188877561484 z)}^{8} + 502404.930773158 {(1.0 x - 2.25500457890024 \cdot 10^{-12} y + 1.88323766493884 \cdot 10^{-9} z)}^{8} + 63398.411897388 {(x - 0.647599822517737 y + 0.843989465785932 z)}^{8} - 153180.375097911 {(x - 0.402229103644926 y + 0.627475858789304 z)}^{8} + 101688.207904785 {(x + 0.511129453461277 y - 0.795587584775735 z)}^{8} - 103622.878332393 {(x + 0.577350269421535 y - 0.872345533336422 z)}^{8} + 63398.4135111495 {(x + 0.647599820452703 y - 0.843989459334063 z)}^{8} - 41092.7531013894 {(x + 0.783778162085557 y - 0.739650840696495 z)}^{8} + 22606.2517808189 {(x + 0.998660464472575 y - 0.57077150098623 z)}^{8} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 2.79227311831127 \cdot 10^{-9} \\ dist (P, Δ) & = & 3.42639808143352e-8 \end{array}

Degree 9 ⟨J U 28⟩ ✖

\begin{array}{rcl} P & = & - 6.47690131545843 \cdot 10^{-14} x^{9} - 0.000894699521756603 x^{8} y - 0.000896140509912845 x^{8} z - 3.09220644734732 \cdot 10^{-10} x^{7} y^{2} - 5.66814054893491 \cdot 10^{-10} x^{7} y z - 2.57068972868567 \cdot 10^{-10} x^{7} z^{2} - 1.51675134076296 x^{6} y^{3} - 4.20115750163369 x^{6} y^{2} z - 3.84483992258021 x^{6} y z^{2} - 1.16042305503365 x^{6} z^{3} - 2.98903753395465 \cdot 10^{-11} x^{5} y^{4} - 6.37583262041529 \cdot 10^{-10} x^{5} y^{3} z + 6.4130497948779 \cdot 10^{-10} x^{5} y^{2} z^{2} + 2.52686110460851 \cdot 10^{-9} x^{5} y z^{3} + 1.27602721429944 \cdot 10^{-9} x^{5} z^{4} + 1.51257607600666 x^{4} y^{5} - 1.40456782361952 x^{4} y^{4} z - 6.40806653727669 x^{4} y^{3} z^{2} + 0.586067739569115 x^{4} y^{2} z^{3} + 7.18995079268125 x^{4} y z^{4} + 3.11292954998475 x^{4} z^{5} + 1.22179986633867 \cdot 10^{-10} x^{3} y^{6} - 2.1265510462967 \cdot 10^{-10} x^{3} y^{5} z - 1.11937042997033 \cdot 10^{-9} x^{3} y^{4} z^{2} + 1.75169667221458 \cdot 10^{-9} x^{3} y^{3} z^{3} + 2.10856613962517 \cdot 10^{-9} x^{3} y^{2} z^{4} - 1.8126185907653 \cdot 10^{-9} x^{3} y z^{5} - 1.38122584014615 \cdot 10^{-9} x^{3} z^{6} - 0.504788491576338 x^{2} y^{7} + 2.32840040407918 x^{2} y^{6} z - 1.28161330874381 x^{2} y^{5} z^{2} - 6.19282709927595 x^{2} y^{4} z^{3} + 4.79330053000986 x^{2} y^{3} z^{4} + 6.22585909895079 x^{2} y^{2} z^{5} - 2.73630046153379 x^{2} y z^{6} - 2.0907856951756 x^{2} z^{7} - 2.61224068948139 \cdot 10^{-11} x y^{8} + 1.62398508872973 \cdot 10^{-10} x y^{7} z - 1.87125656101384 \cdot 10^{-10} x y^{6} z^{2} - 5.99291736462113 \cdot 10^{-10} x y^{5} z^{3} + 1.26840898132158 \cdot 10^{-9} x y^{4} z^{4} + 2.08237515797689 \cdot 10^{-10} x y^{3} z^{5} - 1.33685646372163 \cdot 10^{-9} x y^{2} z^{6} + 1.30498729084796 \cdot 10^{-10} x y z^{7} + 1.79970366202169 \cdot 10^{-10} x z^{8} + 0.0563858433370229 y^{9} - 0.467293133613127 y^{8} z + 1.28161330741715 y^{7} z^{2} - 0.708496733216063 y^{6} z^{3} - 2.39665026294871 y^{5} z^{4} + 3.11292955057919 y^{4} z^{5} + 0.912100152380785 y^{3} z^{6} - 2.09078569526425 y^{2} z^{7} + 1.40425959287793 \cdot 10^{-10} y z^{8} + 0.00705483675959769 z^{9} \\ ≃ & 2380896.82519547 {(- x - 0.729258346206677 y + 0.727781732441915 z)}^{9} - 6947676.05364025 {(- x - 0.619721382928701 y + 0.618725694861009 z)}^{9} + 13220124.7356317 {(- x - 0.577350269174812 y + 0.584631971179082 z)}^{9} - 9609406.44174142 {(- x - 0.536478908104933 y + 0.596825504671969 z)}^{9} + 7241456.37737902 {(- x - 0.44310322926996 y + 0.643168848479633 z)}^{9} + 6821.3739251457 {(- x - 0.268061626727725 y + 0.293770854043093 z)}^{9} + 3112667.81040602 {(- x - 0.0695426730080776 y + 0.829962798402872 z)}^{9} + 9609406.44632266 {(- x + 0.53647890810518 y - 0.596825504526279 z)}^{9} - 13220124.7417725 {(- x + 0.577350269174157 y - 0.584631971033646 z)}^{9} + 2603883.3093554 {(- x + 0.577350269182715 y - 0.576422665048878 z)}^{9} - 2380896.82657527 {(- x + 0.729258346188698 y - 0.72778173227943 z)}^{9} + 380834.337196557 {(- x + 0.96793336044347 y - 0.96637815793565 z)}^{9} - 412.346899407536 {(- x + 0.999790222623675 y - 0.401245079295575 z)}^{9} + 968264.399786432 {(- 0.702257600040002 x - 0.786872848026557 y - z)}^{9} + 589843.548169065 {(- 0.673952454347285 x - y + 0.998446601453092 z)}^{9} - 5671750.2421848 {(- 0.250560639046082 x + y + 0.715839869966238 z)}^{9} + 15415801.7564314 {(- 0.106899414664023 x + y + 0.591377748152394 z)}^{9} + 29862547.5844375 {(- 0.0312057953677463 x - y - 0.526178255824751 z)}^{9} - 29862547.5845394 {(- 0.0312057953401219 x + y + 0.526178255826924 z)}^{9} - 1175698.57508409 {(- 4.40385294436983 \cdot 10^{-11} x + y - 0.998366122433211 z)}^{9} - 48245839.5044969 {(- 1.26525246074838 \cdot 10^{-11} x - y - 0.506306138757093 z)}^{9} + 730564.473970961 {(- 1.04368097832507 \cdot 10^{-11} x - y - 0.463005198034606 z)}^{9} - 9502674.04500366 {(1.22366877958784 \cdot 10^{-11} x + y + 0.499196671225386 z)}^{9} + 15415801.7562901 {(0.106899414699384 x + y + 0.591377748145064 z)}^{9} - 5671750.24205332 {(0.250560639096163 x + y + 0.715839869949062 z)}^{9} - 6824.8260086732 {(0.267836764427584 x - y - 0.29375433989396 z)}^{9} + 6824.82600799203 {(0.267836764437262 x + y + 0.293754339867268 z)}^{9} + 982379.469375189 {(0.329792330460547 x + y - 0.998393240386254 z)}^{9} - 982379.469458807 {(0.329792330545524 x - y + 0.998393240338582 z)}^{9} - 1215553.36513285 {(0.488205894752592 x - y - 0.921365232844725 z)}^{9} + 1215553.3650689 {(0.488205894826941 x + y + 0.921365232811234 z)}^{9} + 589843.548271855 {(0.67395245442239 x - y + 0.998446601355671 z)}^{9} + 968264.400301596 {(0.702257599914497 x - 0.786872847989835 y - z)}^{9} - 1158522.63341402 {(0.867442699517211 x - 0.500818274816218 y - z)}^{9} + 1158522.63260817 {(0.867442699672619 x + 0.500818274840943 y + z)}^{9} - 6947676.05706318 {(x - 0.619721382923902 y + 0.618725694711667 z)}^{9} - 200186.245600638 {(x - 0.57735026918748 y + 0.534632351471496 z)}^{9} + 7241456.38116333 {(x - 0.443103229271273 y + 0.643168848331481 z)}^{9} - 5243069.58284678 {(x - 0.285490253090748 y + 0.722118164843785 z)}^{9} + 6821.37392682709 {(x - 0.268061626751556 y + 0.293770853921731 z)}^{9} + 1292140.98634919 {(x - 0.207960989785132 y - 0.968447332350232 z)}^{9} + 3112667.81254989 {(x - 0.0695426730240845 y + 0.829962798239425 z)}^{9} - 1292140.98737471 {(x + 0.20796098974824 y + 0.968447332170496 z)}^{9} + 5243069.57971777 {(x + 0.285490253085343 y - 0.722118164997361 z)}^{9} + 2603883.30817163 {(x + 0.577350269182815 y - 0.57642266519353 z)}^{9} + 200186.245510888 {(x + 0.57735026918518 y - 0.534632351612168 z)}^{9} + 380834.336903811 {(x + 0.96793336049731 y - 0.966378158133954 z)}^{9} - 412.346899225847 {(x + 0.999790222637554 y - 0.401245079429082 z)}^{9} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 0.695106665257518 \\ dist (P, Δ) & = & 2.45165263090819e-10 \end{array}