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 stopped optimisation when we got the best polynomial with coefficients represented as double (64 bits IEEE 754 representation). When it is not the case, we will continue the optimisation. For the writing as a sum of power of linear forms, we give enough precision for the distance between the two polynomials to be accurate.

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	Remark	Nb. of 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 ✔

This is a M-1 curve, but locally extremal. It is probably not in the same rigid isotopy class than the next one.

\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 ✔

This is a M-1 curve, but locally extremal. It is probably not in the same rigid isotopy class than the previous one.

\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 - 0.0965192797742039 x^{3} y^{2} + 0.53131622091545 x^{3} y z - 0.0965192797742039 x^{3} z^{2} - 0.0965192797742039 x^{2} y^{3} - 3.08382139247772 x^{2} y^{2} z - 3.08382139247772 x^{2} y z^{2} - 0.0965192797742039 x^{2} z^{3} + 0.0298190870015163 x y^{4} + 0.53131622091545 x y^{3} z - 3.08382139247772 x y^{2} z^{2} + 0.53131622091545 x y z^{3} + 0.0298190870015163 x z^{4} - 0.00106873415699291 y^{5} + 0.0298190870015163 y^{4} z - 0.0965192797742039 y^{3} z^{2} - 0.0965192797742039 y^{2} z^{3} + 0.0298190870015163 y z^{4} - 0.00106873415699291 z^{5} \\ ≃ & 19.0067688827383566386 {(- x + 0.060149842239871802922 y - 0.403906402377617521231 z)}^{5} - 4.10506254927064870458 {(- x + 0.151988408912245202572 y - z)}^{5} + 19.0067688827383566386 {(- 0.403906402377617521231 x - y + 0.060149842239871802922 z)}^{5} + 19.0067688827383566386 {(- 0.403906402377617521231 x + 0.060149842239871802922 y - z)}^{5} + 4.10506254927064870458 {(- 0.151988408912245202572 x + y + z)}^{5} - 19.0067688827383566386 {(- 0.060149842239871802922 x + 0.403906402377617521231 y + z)}^{5} - 19.0067688827383566386 {(- 0.060149842239871802922 x + y + 0.403906402377617521231 z)}^{5} - 11.2705241276892411097 {(- 0.0407005400461146206818 x - 0.0407005400461146206818 y + z)}^{5} + 11.2705241276892411097 {(0.0407005400461146206818 x - y + 0.0407005400461146206818 z)}^{5} + 41.481624760977906961 {(0.0740590551607518339134 x + 0.0740590551607518339134 y + z)}^{5} + 41.481624760977906961 {(0.0740590551607518339134 x + y + 0.0740590551607518339134 z)}^{5} - 11.2705241276892411097 {(x - 0.0407005400461146206818 y - 0.0407005400461146206818 z)}^{5} + 41.481624760977906961 {(x + 0.0740590551607518339134 y + 0.0740590551607518339134 z)}^{5} - 19.0067688827383566386 {(x + 0.403906402377617521231 y - 0.060149842239871802922 z)}^{5} + 4.10506254927064870458 {(x + y - 0.151988408912245202572 z)}^{5} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 6.59982960231662876817 \cdot 10^{-18} \\ dist (P, Δ) & = & 0.00249150281902411193057 \end{array}

Degree 5, ⟨𝐽 ∐ 5⟩ ✔

\begin{array}{rcl} P & = & 0.138282576385407 x^{5} + 1.60664479990138 \cdot 10^{-74} x^{4} y + 0.587838979474442 x^{4} z - 1.38282576385407 x^{3} y^{2} - 3.45958159975304 \cdot 10^{-74} x^{3} y z - 2.46107002700762 \cdot 10^{-16} x^{3} z^{2} + 1.47505838426774 \cdot 10^{-74} x^{2} y^{3} + 1.17567795894888 x^{2} y^{2} z - 3.70217191641533 \cdot 10^{-75} x^{2} y z^{2} - 1.09348571906146 x^{2} z^{3} + 0.691412881927033 x y^{4} - 2.47113823740793 \cdot 10^{-74} x y^{3} z - 2.46107002700755 \cdot 10^{-16} x y^{2} z^{2} + 2.7937426435764 \cdot 10^{-74} x y z^{3} + 2.09231774897847 \cdot 10^{-16} x z^{4} - 4.55162698598173 \cdot 10^{-75} y^{5} + 0.587838979474442 y^{4} z + 2.22806310010669 \cdot 10^{-75} y^{3} z^{2} - 1.09348571906146 y^{2} z^{3} + 2.12423438761526 \cdot 10^{-75} y z^{4} + 0.520200768721775 z^{5} \\ ≃ & - 0.29605504806173057544 {(- x - 0.72654252800536085922 y - 0.53504137204818599689 z)}^{5} - 0.71454566055256360844 {(- 0.91062288684363371968 x - 3.607937568104080216 \cdot 10^{-75} y - z)}^{5} + 0.62348654107263842188 {(- 0.8606112651663603633 x - 0.62527068422385940514 y - z)}^{5} + 0.62348654107263842188 {(- 0.8606112651663603633 x + 0.62527068422385940514 y - z)}^{5} + 0.71454566055256397067 {(- 0.73670939092327428073 x - 0.53525070328668537012 y + z)}^{5} - 0.66085414860515297247 {(- 0.32491969623290630207 x - y + 0.98842629975056749903 z)}^{5} - 0.66085414860515297247 {(- 0.32491969623290630207 x + y + 0.98842629975056749903 z)}^{5} - 0.71454566055256373042 {(- 0.28139794750145754851 x - 0.86605383042014205249 y - z)}^{5} + 0.71454566055256373042 {(0.28139794750145754851 x - 0.86605383042014205249 y + z)}^{5} - 0.66468499593665377835 {(0.32491969623290631514 x - y - 0.45513337563457402094 z)}^{5} - 0.66468499593665377835 {(0.32491969623290631514 x + y - 0.45513337563457402094 z)}^{5} - 0.71454566055256397067 {(0.73670939092327428073 x - 0.53525070328668537012 y - z)}^{5} + 0.29605504806173057544 {(x - 0.72654252800536085922 y + 0.53504137204818599689 z)}^{5} + 0.84932516656440315741 {(x + 6.8750384056112487783 \cdot 10^{-75} y - 0.94004927325528403985 z)}^{5} - 0.85424854497517219184 {(x + 8.1446624204447317375 \cdot 10^{-75} y - 0.43285756268067151253 z)}^{5} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.2899195370252784891 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.023035530697453831247 \end{array}

Degree 6, ⟨9 ∐ 1⟨1⟩⟩ ✔

\begin{array}{rcl} P & = & 9.61142335107702 \cdot 10^{-5} x^{6} + 3.94443008515037 \cdot 10^{-10} x^{5} y + 2.67267233665759 \cdot 10^{-10} x^{5} z + 1.89115375851639 x^{4} y^{2} + 2.58331622515706 x^{4} y z + 0.875510022522388 x^{4} z^{2} + 2.66999394915689 \cdot 10^{-10} x^{3} y^{3} - 6.20517332800208 \cdot 10^{-10} x^{3} y^{2} z - 2.05949268851703 \cdot 10^{-9} x^{3} y z^{2} - 9.31641981873725 \cdot 10^{-10} x^{3} z^{3} - 1.26028860154197 x^{2} y^{4} + 1.7222108157218 x^{2} y^{3} z + 1.75102004537569 x^{2} y^{2} z^{2} - 1.04016485661514 x^{2} y z^{3} - 0.646445147783245 x^{2} z^{4} - 1.32826799302121 \cdot 10^{-10} x y^{5} + 6.55961777153812 \cdot 10^{-10} x y^{4} z - 3.68042052082951 \cdot 10^{-10} x y^{3} z^{2} - 9.32752465572185 \cdot 10^{-10} x y^{2} z^{3} + 3.18455509933182 \cdot 10^{-10} x y z^{4} + 1.98003078320781 \cdot 10^{-10} x z^{5} + 0.210192271668139 y^{6} - 0.861105408365832 y^{5} z + 0.875510021681553 y^{4} z^{2} + 0.346721619732493 y^{3} z^{3} - 0.646445147562636 y^{2} z^{4} - 1.37166228257369 \cdot 10^{-10} y z^{5} + 9.61142335107702 \cdot 10^{-5} z^{6} \\ ≃ & 66.93686669425698903529 {(- x - 0.8218932677946685127487 y + 0.6112970526059105521734 z)}^{6} + 29.83886372938646615504 {(- x - 0.5773502693407670262029 y + 0.7755416011207467205052 z)}^{6} - 161.4834524074586203881 {(- x - 0.5773502693198680564291 y + 0.6680498092570660452873 z)}^{6} + 211.9380952162161431169 {(- x - 0.3755455622667985054818 y + 0.5044618876121104498163 z)}^{6} + 364.9834573493371747307 {(- x - 2.669008966417179101189 \cdot 10^{-13} y + 1.530793450735262160484 \cdot 10^{-10} z)}^{6} - 310.3206708905372230308 {(- x + 0.1794164579134390463943 y - 0.2584626453433536742282 z)}^{6} - 161.483452605925767409 {(- x + 0.5773502692010713005547 y - 0.6680498088140657390602 z)}^{6} + 29.83886377195106663602 {(- x + 0.577350269202969934971 y - 0.7755416006302053481315 z)}^{6} - 67.92753763377424356878 {(- 0.844215781348180053884 x - y + 0.3329343390973398906134 z)}^{6} + 70.72915848987851912085 {(- 1.025471157509130158372 \cdot 10^{-10} x - y - 0.6716387282567188091039 z)}^{6} - 382.7755909860688652925 {(- 8.829686423469717273501 \cdot 10^{-11} x - y - 0.5785481058127065480823 z)}^{6} + 3.183370227680662699357 {(1.530793450735540212018 \cdot 10^{-10} x - 1.060454037763545767665 \cdot 10^{-10} y + z)}^{6} + 290.2376199205781421498 {(0.1658457998032896828955 x - y - 0.478708252551350989984 z)}^{6} + 290.2376199207323084534 {(0.1658457999493018905309 x + y + 0.4787082525005334776673 z)}^{6} - 236.5004226472435980178 {(0.3605824654230248522674 x - y - 0.2704337660992902758391 z)}^{6} - 236.5004226475167270111 {(0.3605824655052172933009 x + y + 0.2704337659888427675485 z)}^{6} + 153.9773960693955920018 {(0.5773502691892671343616 x + y + 1.766848541306840039539 \cdot 10^{-11} z)}^{6} + 153.9773960691108656769 {(0.5773502691899788700535 x - y - 1.94429287584244143221 \cdot 10^{-10} z)}^{6} - 67.92753763359057678674 {(0.8442157814510250374912 x - y + 0.332934338839025927607 z)}^{6} + 66.93686676959816746622 {(x - 0.8218932676399535888585 y + 0.6112970521850770384516 z)}^{6} + 211.9380954128682341349 {(x - 0.3755455622081881381574 y + 0.5044618872279388184746 z)}^{6} - 310.3206707430237639868 {(x + 0.1794164579281873606829 y - 0.2584626456699894184131 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 3.701534591526215018437 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.00009611423351072587523320 \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} + 0.000589006101698443 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} \\ ≃ & 87.66646864564133967407 {(- x - 0.1507293597227276952201 y - 0.1157839790016475252289 z)}^{6} + 265.6957559909633296016 {(- x + 0.065650241518947071828 y - 0.0859866326669048964606 z)}^{6} - 153.7999933798103593101 {(- x + 0.3279036352493385149154 y - 0.03357610317218884702966 z)}^{6} + 56.51911811773487533917 {(- x + 0.7048539152754089039994 y + 0.002037252838309579270864 z)}^{6} - 50.83277840913819544554 {(- 0.7755717129177099727248 x + y + 0.04441005916920915745317 z)}^{6} - 124.6938202985261106282 {(- 0.2149458091187146144034 x - 0.3078411813492471006455 y + z)}^{6} + 83.66722354267773963712 {(- 0.1954058467210668898676 x + y + 0.1191397881834638853994 z)}^{6} + 127.02202342069828286 {(- 0.1951164913941355558577 x + y + 0.4034946388763416214009 z)}^{6} - 107.2131512627209183429 {(- 0.1472917695691492199013 x - 0.02571699358497896138242 y + z)}^{6} + 22.36546402489446321938 {(- 0.1292507806402819707198 x + 0.671894154008624656853 y + z)}^{6} + 406.7913764011825171958 {(- 0.09415465369625705064322 x - 0.1364706990341131093822 y + z)}^{6} - 121.2378219049532003115 {(- 0.0008313540981224980538622 x - 0.004069789319835573878709 y + z)}^{6} - 106.9836117662367414535 {(0.02802591735275618142236 x - 0.1493366314922560001229 y + z)}^{6} - 28.15423506287319001111 {(0.1751266439942546998774 x - y - 0.7963553267184028747649 z)}^{6} - 301.3127160309538666799 {(0.2514791432050271865961 x - y - 0.1823871641707819966009 z)}^{6} + 22.00298558166379189693 {(0.3299484396081005613924 x + 0.4709992882946424591097 y - z)}^{6} + 160.6497480531579803449 {(0.4325505542742250263193 x - y - 0.09660458117281437004966 z)}^{6} + 22.14541879693895914916 {(0.6800810289776988237387 x + 0.1023913279096786582978 y + z)}^{6} - 353.1992244036750545862 {(x + 0.09638920334209254978348 y + 0.1750846373868174072439 z)}^{6} + 133.6307590682774476811 {(x + 0.1503574351733348949438 y + 0.3976667330107091116902 z)}^{6} - 28.63718263211169353803 {(x + 0.1692261177187930400198 y + 0.7917080778383553338454 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 8.970496853187705608646 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.0002488484400222492719633 \end{array}

Degree 6, ⟨2 ∐ 1⟨6⟩⟩ ✔

\begin{array}{rcl} P & = & - 0.396663870419484 x^{6} + 0.216401368186008 x^{5} y - 0.0109419425766376 x^{5} z - 0.817302526519895 x^{4} y^{2} + 0.480727148018004 x^{4} y z + 1.27206642924796 x^{4} z^{2} + 0.0508999223807361 x^{3} y^{3} - 0.175349776088968 x^{3} y^{2} z - 0.45114996633883 x^{3} y z^{2} + 0.0232569761493416 x^{3} z^{3} - 0.163029808913808 x^{2} y^{4} + 1.04638338230139 x^{2} y^{3} z + 1.511954759629 x^{2} y^{2} z^{2} - 1.04470250584592 x^{2} y z^{3} - 1.35783369450648 x^{2} z^{4} - 0.234906246892977 x y^{5} - 0.166627696307577 x y^{4} z + 0.0221136877021396 x y^{3} z^{2} + 0.181835573925925 x y^{2} z^{3} + 0.235225289191958 x y z^{4} - 0.0123739559325354 x z^{5} + 0.194998705698053 y^{6} + 0.566460762177821 y^{5} z - 0.00753890287955779 y^{4} z^{2} - 1.13007440317115 y^{3} z^{3} - 0.670730783713313 y^{2} z^{4} + 0.566435465179922 y z^{5} + 0.482446648615305 z^{6} \\ ≃ & - 17235.78104656469777543423 {(- x - 0.2851841795462851317245997 y - 0.9466525988606789114096973 z)}^{6} + 5899.186158589405233457127 {(- x - 0.1537138960988787182772355 y - 0.9312014361754190116497741 z)}^{6} + 11386.00572191097291812647 {(- x + 0.06854216570187982392142359 y + 0.9683731905992067009047867 z)}^{6} + 15282.4491096445364669982 {(- 0.9548768722084107601120882 x + 0.3723098942710990744928516 y + z)}^{6} + 8332.793102260962060854576 {(- 0.9524616191599307072857394 x - 0.6088550473196984923098211 y - z)}^{6} - 12283.77806001963368559242 {(- 0.7738609928896310583102553 x + 0.6246203299450855773701952 y + z)}^{6} - 7005.341879379927003523533 {(- 0.6889837205115018911078436 x - 0.8815560403531281036540565 y - z)}^{6} + 17902.62670914677740190232 {(- 0.6691231240173715846583447 x + 0.6996944212365424607237077 y - z)}^{6} + 10250.59897142271826086176 {(- 0.4826816020788401459189845 x + 0.854600981803520362405341 y + z)}^{6} - 8637.51376327745840008091 {(- 0.107499146744896281306308 x + y + 0.9985571075852862833919339 z)}^{6} + 8228.001363510637192621058 {(0.3006376798478189801623951 x + y + 0.9806010842921650477007078 z)}^{6} - 1178.711577443344821620178 {(0.4249775144365260467121012 x - y + 0.9217325979782654438952338 z)}^{6} + 3450.569184473350749611623 {(0.5069223617669937508654079 x - 0.9772348024447241086665174 y + z)}^{6} - 9242.362437792990698670126 {(0.5827266929349213753137055 x - 0.8368117394448596856598343 y + z)}^{6} - 5161.02207249004509078564 {(0.6585249964909451602270775 x - 0.6822493671684331260200468 y + z)}^{6} - 7176.854019404111380260462 {(0.7710661150851699824726115 x - 0.5732827031343664668931111 y + z)}^{6} - 24264.08470310886763988466 {(x - 0.1699662094180328912497972 y - 0.9706045077995759175487506 z)}^{6} + 6651.727839634880721063516 {(x - 0.1176627272002428425493434 y - 0.9485086878979272230989764 z)}^{6} - 2784.698772054462158593742 {(x + 0.001988443459982864912715836 y - 0.9833693350774055594212608 z)}^{6} + 6426.084877326416573570233 {(x + 0.1137951414512798495084919 y + 0.9509181895938248086911975 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 4.574136275902520157495068 \cdot 10^{-8} \\ dist (P, Δ) & = & 0.0000006564983739755009012775323 \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} - 8.65065554707418 \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} + 5.80365292711414 \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} \\ ≃ & 1850555.33531871767872576778 {(- 0.987851014885905556112132088 x - 0.106139848843755440937596419 y + z)}^{6} - 2374262.60452713702992949187 {(- 0.980532959723679492697858044 x - 0.260438506345591706450492045 y + z)}^{6} + 574945.970085544882459069942 {(- 0.977062516154462552470284354 x + 0.30143858863936072045498432 y - z)}^{6} + 1019052.94417849084735622131 {(- 0.967090225033569233549743316 x + 0.327324104803203692374806041 y - z)}^{6} - 1494724.01187274258520413641 {(- 0.956483498679631273516771421 x - 0.132187301948753472369365336 y - z)}^{6} - 251102.793085081304267999897 {(- 0.954196444521114759678632058 x + 0.368561915919026973492572624 y - z)}^{6} + 1250246.14653578283951981878 {(- 0.848183630165365173823307262 x + 0.411728060938907547077948515 y + z)}^{6} - 1080725.46149209752927984505 {(- 0.643485826161730270419067755 x + 0.675414091581846399692258685 y + z)}^{6} - 1080726.14787067089074780741 {(- 0.643485624785716935678759652 x - 0.675413990329890005506787027 y - z)}^{6} + 976119.653393515255634480123 {(- 0.349228101934146883553936377 x - 0.865220308762646925545633402 y - z)}^{6} - 940224.535310894976997853716 {(0.0000000606365882411177047991768419 x - 0.934658609081793445505550325 y - z)}^{6} - 5257.9847457654571954799101 {(0.0000000686626322520361584348302571 x + y - 0.553685855372668448014504131 z)}^{6} + 5465.53453057965344065469568 {(0.0000000690898645192630566318318138 x + y - 0.558880262824078914522898743 z)}^{6} + 976119.316942738973013719322 {(0.349228246480645719652314676 x - 0.865220374617217823073663835 y - z)}^{6} + 1250247.19316969994434407125 {(0.848183366367290820176171714 x + 0.411727964268361509828039234 y + z)}^{6} - 251102.556602864720486955712 {(0.954196775835579215055329061 x + 0.36856192964203889893423908 y - z)}^{6} - 1494722.60080573586348237709 {(0.956483807555469958027849788 x - 0.132187366980091253029061461 y - z)}^{6} + 1019051.97149199073769497642 {(0.967090558515994396613033063 x + 0.327324112151395466322696239 y - z)}^{6} + 574945.415640468883848205288 {(0.977062851629009230847908183 x + 0.301438591902903056612236266 y - z)}^{6} - 2374264.90226438229688327137 {(0.980532625027766375790344971 x - 0.260438509683555456262138873 y + z)}^{6} + 1850557.13959460227201631055 {(0.987850684956087150871777874 x - 0.106139877279932824579427874 y + z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 6.98840116117683748992644576 \cdot 10^{-17} \\ dist (P, Δ) & = & 5.82146440172127668159727107e-9 \end{array}

Degree 6, ⟨5 ∐ 1⟨5⟩⟩ ✔

\begin{array}{rcl} P & = & - 0.00132990000822424 y^{6} - 0.0616200461333061 y^{5} (0.707106781186548 x - 0.707106781186548 z) + 0.00104567643964428 y^{5} (0.707106781186548 x + 0.707106781186548 z) + 0.0423766780436503 y^{4} (0.707106781186548 x - 0.707106781186548 z) (0.707106781186548 x + 0.707106781186548 z) - 0.00223803203245656 y^{4} {(x - z)}^{2} + 0.000244121531585368 y^{4} {(x + z)}^{2} + 0.0285392824519228 y^{3} (0.707106781186548 x - 0.707106781186548 z) {(x + z)}^{2} - 0.129254584962048 y^{3} (0.707106781186548 x + 0.707106781186548 z) {(x - z)}^{2} + 0.00038975448293896 y^{3} {(x - z)}^{3} + 0.000124842074475883 y^{3} {(x + z)}^{3} + 0.0124919547764453 y^{2} (0.707106781186548 x - 0.707106781186548 z) {(x + z)}^{3} - 0.00354416617249993 y^{2} (0.707106781186548 x + 0.707106781186548 z) {(x - z)}^{3} + 2.42636930582478 \cdot 10^{-6} y^{2} {(x - z)}^{4} + 0.394230632890865 y^{2} {(x - z)}^{2} {(x + z)}^{2} + 4.40611823967971 \cdot 10^{-6} y^{2} {(x + z)}^{4} + 0.000427153819177914 y (0.707106781186548 x - 0.707106781186548 z) {(x + z)}^{4} + 0.000189092943319493 y (0.707106781186548 x + 0.707106781186548 z) {(x - z)}^{4} - 2.47042452224817 \cdot 10^{-7} y {(x - z)}^{5} - 0.0165983876059155 y {(x - z)}^{3} {(x + z)}^{2} + 0.17267379164541 y {(x - z)}^{2} {(x + z)}^{3} - 3.63005780728573 \cdot 10^{-8} y {(x + z)}^{5} - 1.66047194284044 \cdot 10^{-6} (0.707106781186548 x - 0.707106781186548 z) {(x + z)}^{5} + 2.25364436888175 \cdot 10^{-5} (0.707106781186548 x + 0.707106781186548 z) {(x - z)}^{5} - 1.61812343280988 \cdot 10^{-8} {(x - z)}^{6} - 0.00515590206634466 {(x - z)}^{4} {(x + z)}^{2} + 0.54606727014632 {(x - z)}^{3} {(x + z)}^{3} + 0.00518676318548151 {(x - z)}^{2} {(x + z)}^{4} + 6.84766574026013 \cdot 10^{-11} {(x + z)}^{6} \\ ≃ & - 167462140.32117677592582168358 {(- x - 0.084181782802766799879916440971 y + 0.9974852878572141014132616306 z)}^{6} + 55055739.950197241878158552097 {(- x + 0.11436448110736755719389432427 y - 0.99621286273005385614797831235 z)}^{6} - 8280847.7360865355943936287597 {(- 0.98163738824615463354150158345 x - 0.60882528426630053495949096454 y - z)}^{6} - 70142245.451291447955336302964 {(- 0.97970771131629796329739746392 x - 0.25373920803242602980544884241 y + z)}^{6} + 1862150.5517169036308242730103 {(- 0.97900135498655134140733908126 x - 0.69633626062807561577225181446 y - z)}^{6} + 38643974.717445314099713645182 {(- 0.95948620070216123985588633687 x - 0.35162794432344637885852215466 y + z)}^{6} - 16838923.725209089554815053206 {(- 0.93766326999738530010922603652 x - 0.4378096613750892074085343554 y + z)}^{6} + 4147402.5740593280412741730889 {(- 0.92048204998039849067433308197 x - 0.49709751059120110470377190439 y + z)}^{6} + 21347925.565222100643432932475 {(0.9855386550287656612923735643 x + 0.48252183183334361578684396936 y + z)}^{6} - 43159828.919499375098301626191 {(0.99001295317263818780202295355 x + 0.33775865990516908195210552143 y + z)}^{6} + 111921653.01876046437775497007 {(0.99434630020897723797598866862 x + 0.16004940694737110743775623358 y - z)}^{6} + 74322732.020390615756274729543 {(0.99438795537665267011450475627 x + 0.19308917815395283728707530486 y + z)}^{6} - 113859450.91621796480184305943 {(0.99824272698691113713069373025 x + 0.064154269934946315798612237609 y + z)}^{6} - 24090665.934844123380331183369 {(x - 0.14684046078429308887693881534 y + 0.99261003847957430150236304041 z)}^{6} + 97398696.995193124792336316032 {(x - 0.12766463086945173303767093958 y + 0.99439191838132707435768756648 z)}^{6} - 223256453.98552784480523301915 {(x - 0.098887547990085134117045477253 y + 0.99631388103306565114077896551 z)}^{6} + 162547806.18480818682691216728 {(x - 0.036174038019478416551774100403 y + 0.99860281081077831636305369513 z)}^{6} + 26165484.285894181499718319214 {(x - 0.0025482147412359829268913217337 y - 0.99464857619524777612111880882 z)}^{6} - 105380059.31395465914766802239 {(x + 0.01303162033943960789971042767 y - 0.99410995347702369415069977405 z)}^{6} - 59617163.016008478383347853551 {(x + 0.024203051084351045710478074511 y - 0.99323200426818165404144889144 z)}^{6} + 238909354.14469809100651628747 {(x + 0.035842006661433919634091021569 y - 0.99428263984181616711432178559 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.0423792679257869683000241876 \cdot 10^{-14} \\ dist (P, Δ) & = & 7.3832379799858445485227115341e-11 \end{array}

Degree 7, ⟨𝐽 ∐ 15⟩ ✔

\begin{array}{rcl} P & = & - 1.74415402874978 \cdot 10^{-10} x^{7} + 0.193985061996207 x^{6} y - 0.0674698195570194 x^{6} z + 1.47457241517702 \cdot 10^{-8} x^{5} y^{2} - 3.34288587749802 \cdot 10^{-8} x^{5} y z + 1.01491178382314 \cdot 10^{-8} x^{5} z^{2} + 3.19653872230425 x^{4} y^{3} - 0.537105063338043 x^{4} y^{2} z - 1.12612133763269 x^{4} y z^{2} + 0.322178912498801 x^{4} z^{3} - 6.16682520044354 \cdot 10^{-8} x^{3} y^{4} - 2.28770197209962 \cdot 10^{-7} x^{3} y^{3} z + 1.98897825628962 \cdot 10^{-8} x^{3} y^{2} z^{2} + 1.10575615971594 \cdot 10^{-7} x^{3} y z^{3} - 3.04774765965544 \cdot 10^{-8} x^{3} z^{4} - 0.412696946812073 x^{2} y^{5} - 2.87922246275859 x^{2} y^{4} z - 4.69434441310397 x^{2} y^{3} z^{2} + 0.79744414055892 x^{2} y^{2} z^{3} + 1.81877730525204 x^{2} y z^{4} - 0.487300026163997 x^{2} z^{5} + 3.89755112946603 \cdot 10^{-9} x y^{6} + 4.20794560514053 \cdot 10^{-8} x y^{5} z + 1.50063760088956 \cdot 10^{-7} x y^{4} z^{2} + 1.73813500346809 \cdot 10^{-7} x y^{3} z^{3} - 3.22693680864886 \cdot 10^{-8} x y^{2} z^{4} - 7.91808693362777 \cdot 10^{-8} x y z^{5} + 2.17225891352177 \cdot 10^{-8} x z^{6} + 0.0132679769536452 y^{7} + 0.187064674904409 y^{6} z + 0.978383145670196 y^{5} z^{2} + 2.24737285428712 y^{4} z^{3} + 1.84740166166103 y^{3} z^{4} - 0.46881063416948 y^{2} z^{5} - 0.81032990850427 y z^{6} + 0.222620135268147 z^{7} \\ ≃ & 737.583632249725877922106 {(- x - 0.7430393317874477789192 y - 0.8783032079440896723802 z)}^{7} - 522.166745110871534332099 {(- x - 0.345201971887186757688792 y - 0.992400308938238450847721 z)}^{7} + 13.5338736804438272108341 {(- x - 0.032544020329245999475341 y + 0.702663067722345839900194 z)}^{7} + 1392.08351380700445739413 {(- x + 0.496890579078654594886053 y + 0.950997197283469082170708 z)}^{7} - 737.583529677060280996719 {(- x + 0.743039351238252417263789 y + 0.878303244045156127144227 z)}^{7} + 2712.37399705287891175064 {(- 0.810561602964861236527234 x + 0.347844590607047120409006 y + z)}^{7} - 2776.33933459518796115043 {(- 0.571963633287207716347078 x + y + 0.350841178780400211176522 z)}^{7} - 4023.11923601468166064837 {(- 0.558694059828629659752311 x + 0.344795883897282020108047 y + z)}^{7} - 7031.52375360478748348248 {(- 0.355160749944217022218366 x - y - 0.112982103724865975264171 z)}^{7} + 5877.09335304817717709326 {(- 0.278905248244801235366719 x + 0.347843056650947065641545 y + z)}^{7} + 12224.3119313938492766935 {(- 0.192898240460801982842019 x - y + 0.0670226346514183292531303 z)}^{7} - 17593.4136549058751564525 {(- 0.0754484589178065130048167 x - y + 0.19095631764005904426417 z)}^{7} + 6816.4262086221959207539 {(- 1.01449680877731271488031 \cdot 10^{-8} x - 0.349255322568968955579317 y - z)}^{7} + 5486.17016256516551167844 {(- 2.00869668293423156226093 \cdot 10^{-10} x + y - 0.272939103913952872448434 z)}^{7} + 25906.5374312034528745542 {(- 1.01336532310860083546528 \cdot 10^{-10} x - y + 0.240534781256189021570316 z)}^{7} - 15.7168985315477461884406 {(3.78709129858557151628194 \cdot 10^{-9} x - y + 0.657474836793920856895284 z)}^{7} - 17593.4136984766606689627 {(0.0754484577636922460966513 x - y + 0.190956316165219462095674 z)}^{7} + 12224.3120087950311132039 {(0.192898236847259869017996 x - y + 0.0670226309928145783016208 z)}^{7} + 5877.09313903143764815481 {(0.278905269979036241087271 x + 0.347843057152659693078279 y + z)}^{7} - 7031.52383557750175703877 {(0.355160742556181711798671 x - y - 0.112982110161238846998282 z)}^{7} - 4023.1189425435335787873 {(0.558694085919739795318808 x + 0.344795884870542444061626 y + z)}^{7} - 2776.33928247141214495846 {(0.571963646054381875496642 x + y + 0.350841169052981428518166 z)}^{7} + 2712.37370999858769567625 {(0.810561635502876765418596 x + 0.347844592065160428261279 y + z)}^{7} - 661.484304215640092329288 {(0.878865467022532596709758 x - y - 0.682048508368015631468337 z)}^{7} + 661.484285133070808973137 {(0.878865488055371175762438 x + y + 0.682048494786057242573682 z)}^{7} + 2379.42098829348197721524 {(0.973785922083014070403365 x - 0.379977871411113311314881 y - z)}^{7} - 2379.42068576712383386505 {(0.973785960204145883455996 x + 0.379977873746494947744039 y + z)}^{7} - 522.166671535363017454661 {(x - 0.345201983524998599589661 y - 0.992400347566657150202895 z)}^{7} + 13.5338749076322683399112 {(x - 0.0325440245968627515113304 y + 0.702663039968101975813966 z)}^{7} + 1392.08370936341937722393 {(x + 0.496890564417769086891484 y + 0.950997159546439708772479 z)}^{7} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 9.3661690299577384587753 \cdot 10^{-17} \\ dist (P, Δ) & = & 0.00000238287955327058457642401 \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} + 1.74698495308752 \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} + 5.12029721743252 \cdot 10^{-9} y z^{7} - 0.215868035709408 z^{8} \\ ≃ & - 41092.753101152946135664631 {(- x - 0.78377816208381896545425688 y + 0.73965084069786299859205662 z)}^{8} + 101688.20790401539286398679 {(- x - 0.51112945346182100011645246 y + 0.79558758477653398332032514 z)}^{8} + 272236.89420214674728393621 {(- x - 0.26866765768780414913863019 y + 0.41818887756115697704011554 z)}^{8} + 272236.89076910931133645488 {(- x + 0.2686676581158176183230246 y - 0.41818888198682753313923478 z)}^{8} - 153180.37509889329580389426 {(- x + 0.40222910364351236450615708 y - 0.62747585878708633683521774 z)}^{8} + 101688.20546442076818924378 {(- x + 0.51112945499913986400789293 y - 0.79558759092887005607543752 z)}^{8} + 573.82086707385807197495238 {(- 0.91247685013667526444729847 x + 0.52681875750222005432955982 y - z)}^{8} + 158964.06012228293637267103 {(- 0.57735026918661909382640946 x + y + 2.6918972126833281548672355 \cdot 10^{-10} z)}^{8} - 243453.63867355946852148606 {(- 0.41304212435818333218493726 x + y + 0.22113039066831765695352774 z)}^{8} + 273020.24625132097104870222 {(- 0.26723098646731015913584743 x + y + 0.41803870815880216665364816 z)}^{8} - 257616.79657405273269001908 {(- 0.14211761058350007712286153 x + y + 0.58799771807245375231205929 z)}^{8} - 257616.79657273177327057023 {(- 0.14211760837342227664952697 x - y - 0.58799771860811310049052458 z)}^{8} + 254650.19728016590324178957 {(- 0.05113178954334642829357377 x + y + 0.70933897257945121066257308 z)}^{8} + 254650.19727969611471613688 {(- 0.051131786876160487588622976 x - y - 0.70933897277220140651781269 z)}^{8} + 62474.945637616317852239657 {(- 1.4634015991045443396840104 \cdot 10^{-9} x + y + 0.77826427910536737490115777 z)}^{8} - 327499.46033021113115662868 {(1.4204809541986528049259963 \cdot 10^{-9} x - y - 0.75547339842913188206890986 z)}^{8} + 871.57773423290117543935187 {(1.78511270603536564397396 \cdot 10^{-9} x - y - 0.9490930135373968937752746 z)}^{8} - 20.09668559988582227673035 {(1.8849436086770641560948949 \cdot 10^{-9} x + 0.75651455491123575518210028 y - z)}^{8} + 273020.24624868858806716354 {(0.26723098489760975574256869 x + y + 0.41803870916582490776881038 z)}^{8} - 243453.63866993137877067856 {(0.41304212353058060192301557 x + y + 0.22113039222444255390801505 z)}^{8} + 158964.06011897158149883914 {(0.57735026919263243420661713 x + y + 2.4437652668362515602945954 \cdot 10^{-9} z)}^{8} - 20.096686045729347223753604 {(0.65516081801184222361669681 x + 0.37825727437047294989026441 y + z)}^{8} - 20.096685648997123923518326 {(0.65516082339331728810350926 x - 0.37825727530683318645972632 y - z)}^{8} - 71304.952265795933478619423 {(0.76891747478018995264070886 x - y + 0.25781637885680186933046808 z)}^{8} - 71304.95226381774881049035 {(0.76891747575842545799966033 x + y - 0.25781637596158723009903623 z)}^{8} + 573.82085129689887742654548 {(0.91247685704154667331997448 x + 0.52681875930868684148433881 y - z)}^{8} + 22606.251390899473160342848 {(x - 0.9986604666328049374294719 y + 0.57077150597894813267481413 z)}^{8} - 41092.752184156171237602758 {(x - 0.78377816427460879780713969 y + 0.73965084652752889889957734 z)}^{8} + 63398.411897559632777194184 {(x - 0.64759982251797063795020648 y + 0.84398946578576729645964656 z)}^{8} + 19767.46316725915625831158 {(x - 0.57735027138875186166523813 y + 0.89866218219140255852657055 z)}^{8} - 103622.87560667255350504408 {(x - 0.57735027132444016546814445 y + 0.87234553997136114343871012 z)}^{8} - 426329.31707296075435436071 {(x - 0.13267026916534226112727953 y + 0.20617325123822298269600597 z)}^{8} + 502404.93075161207419456861 {(x - 2.2550045585050086749380026 \cdot 10^{-12} y + 1.8832376649071639317685996 \cdot 10^{-9} z)}^{8} - 426329.31972351766144087686 {(x + 0.13267026905772824209846286 y - 0.20617324731152118406435103 z)}^{8} - 153180.37799729442944874343 {(x + 0.4022291026876555552377843 y - 0.62747585353651365246914212 z)}^{8} - 103622.87833259402512686151 {(x + 0.57735026942144554479490381 y - 0.87234553333637658745334096 z)}^{8} + 19767.463702940459802742603 {(x + 0.57735026942852971433156211 y - 0.89866217538080545153828046 z)}^{8} + 63398.413511321174950951057 {(x + 0.64759982045293647284234237 y - 0.84398945933389796666359291 z)}^{8} + 22606.251780504791963144767 {(x + 0.99866046447687993375220631 y - 0.57077150098285932219238387 z)}^{8} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 7.1837734110860747196532771 \cdot 10^{-17} \\ dist (P, Δ) & = & 3.4263980840656347605093331e-8 \end{array}

Degree 8, ⟨19⟩ ✔

\begin{array}{rcl} P & = & 0.0495589143528351 x^{8} - 0.00188018680131159 x^{7} y - 0.000795119640932293 x^{7} z - 0.98222967758031 x^{6} y^{2} - 0.512485805811381 x^{6} y z - 0.387887731197622 x^{6} z^{2} + 0.0196278020389989 x^{5} y^{3} + 0.0177437997933578 x^{5} y^{2} z + 0.0131094554566617 x^{5} y z^{2} + 0.00388115481719391 x^{5} z^{3} + 4.81286215768501 x^{4} y^{4} + 4.37041111569147 x^{4} y^{3} z + 5.46367682835569 x^{4} y^{2} z^{2} + 2.26805542914645 x^{4} y z^{3} + 0.96347855083981 x^{4} z^{4} - 0.0115396380424817 x^{3} y^{5} - 0.00891660576928047 x^{3} y^{4} z - 0.0238949884583324 x^{3} y^{3} z^{2} - 0.0108324040428585 x^{3} y^{2} z^{3} - 0.0125477647548797 x^{3} y z^{4} - 0.00487482221080735 x^{3} z^{5} + 0.245163497293747 x^{2} y^{6} + 3.32420020860353 x^{2} y^{5} z - 1.75134056243491 x^{2} y^{4} z^{2} + 2.90577734180137 x^{2} y^{3} z^{3} - 1.78664997060123 x^{2} y^{2} z^{4} - 2.14515883780769 x^{2} y z^{5} - 0.882233333049585 x^{2} z^{6} - 0.000352126771039669 x y^{7} - 0.00481778746941958 x y^{6} z - 0.000622356910905359 x y^{5} z^{2} - 0.00303919507158106 x y^{4} z^{3} + 0.00499500693115176 x y^{3} z^{4} - 0.000908018084730323 x y^{2} z^{5} + 0.0030737225764952 x y z^{6} + 0.00177576523890042 x z^{7} - 0.000593764825451007 y^{8} + 0.00802584243368985 y^{7} z + 0.121879421524613 y^{6} z^{2} - 1.1278120374435 y^{5} z^{3} + 2.6700988140946 y^{4} z^{4} - 1.70424194371066 y^{3} z^{5} - 0.740965215107721 y^{2} z^{6} + 0.511633629359521 y z^{7} + 0.262270902028137 z^{8} \\ ≃ & 1.390023128461474579 {(- x - 0.514864672901870670517 y + 0.694792149280820947404 z)}^{8} - 5.76856376174915765797 {(- x - 0.367651952864542142082 y + 0.449029247862737071282 z)}^{8} + 11.0189889580600322248 {(- x - 0.348221060805872479425 y + 0.162547771489192290354 z)}^{8} + 5.28914333657159287355 {(- x - 0.0623240371443920724944 y + 0.863300220608303397447 z)}^{8} - 13.6179238115978697977 {(- x - 0.0133017772624531407553 y + 0.580884258432679159669 z)}^{8} - 12.9799982438379493272 {(- x + 0.0147305009770848275137 y - 0.57992073239483549944 z)}^{8} - 15.0428544729854527006 {(- x + 0.286420392192215051987 y + 0.106533249699119379954 z)}^{8} - 8.21674557054554787186 {(- 0.832142048755301378634 x - 0.0523334213113831989111 y - z)}^{8} - 8.11552529248380784848 {(- 0.830832358180963667038 x + 0.053493992183472768816 y + z)}^{8} + 3.43594288028844556097 {(- 0.693008211955235835485 x - y + 0.596367698528421295515 z)}^{8} + 3.5179481547302908655 {(- 0.691321266081178478422 x + y - 0.595271246521781144432 z)}^{8} + 9.12166752574641419074 {(- 0.627558131122870499815 x - 0.333099558503366360046 y - z)}^{8} + 9.03462017052236159039 {(- 0.626063742643032917838 x + 0.33384616536963884814 y + z)}^{8} - 6.80361606701100434313 {(- 0.42602969155936136302 x + 0.616319883065497222444 y + z)}^{8} - 18.7234703598625914949 {(- 0.381994900341736507673 x - y + 0.317287446742690586326 z)}^{8} - 18.9774784807734199622 {(- 0.380583277447180849589 x + y - 0.316835905923849674503 z)}^{8} + 3.69781507743267734732 {(- 0.215922762999589999033 x - 0.895930219341788753721 y - z)}^{8} - 5.50865556312886726439 {(- 0.0010471814359080330952 x - y - 0.904757201368224393515 z)}^{8} - 100.113503296839096484 {(- 0.000708485063682819988933 x - y + 0.0238613644964277896563 z)}^{8} + 33.9217223339916592146 {(0.000769903958928277064165 x + y + 0.144533490296711020538 z)}^{8} - 8.8617659631969015615 {(0.000852723761545914622879 x + y + 0.371604143066634607299 z)}^{8} + 5.01134119110655147787 {(0.000949962549676449763213 x + y + 0.638207955168146082825 z)}^{8} + 52.1382396941048274585 {(0.160693766253681369832 x - y + 0.129247334735093325364 z)}^{8} + 51.8119431496453608733 {(0.162071164685063605015 x + y - 0.129394992845831119458 z)}^{8} + 3.67917075679684490081 {(0.213874391306113097406 x - 0.896098005017882744826 y - z)}^{8} - 6.85606322258123997565 {(0.427775829028201283447 x + 0.615899406602238563444 y + z)}^{8} - 0.168158090550749774952 {(0.715322446548715959756 x - 0.372214115513973854825 y + z)}^{8} + 2.19386791269369689742 {(0.828811689671995075658 x + 0.195574543656054599001 y - z)}^{8} + 2.37200760180158879011 {(0.829761350398868711896 x - 0.19688275752418034993 y + z)}^{8} + 0.0713041294826755666898 {(0.936821208170425277126 x - 0.570529403619022106307 y + z)}^{8} - 0.980972440487592667718 {(x - 0.834524192744189608891 y + 0.869136844752565805157 z)}^{8} + 1.32947040636650824212 {(x - 0.516419196823238992133 y + 0.694224511242661041957 z)}^{8} - 5.07900564980871701715 {(x - 0.369160421308773292273 y + 0.448390027651836735607 z)}^{8} + 10.5872881566921501478 {(x - 0.349788852778525302644 y + 0.161880319012975976822 z)}^{8} + 2.53494813346077842487 {(x - 0.220157481488407112402 y + 0.590156923216909136977 z)}^{8} + 15.583818312096740699 {(x - 0.159136773462445597936 y - 0.351210389544920435209 z)}^{8} + 5.25385219757797448039 {(x - 0.0637244489218174766948 y + 0.862105839372452686507 z)}^{8} + 15.2447221886636640524 {(x + 0.157625892024054788088 y + 0.350311493330204895941 z)}^{8} + 3.25367506160861802164 {(x + 0.218748564203778468505 y - 0.590954896662565830591 z)}^{8} - 15.1015917993935178387 {(x + 0.284846874561262385414 y + 0.10575208127168468563 z)}^{8} - 0.953137039630366940463 {(x + 0.832621710691360088755 y - 0.869378162435763103742 z)}^{8} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.37922513912681184877 \cdot 10^{-16} \\ dist (P, Δ) & = & 0.000698170370131225757846 \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}