Exceptional polynomials

This page gives examples of 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 the polynomial 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) (marked as 🟢). When it is not the case, we will continue the optimisation (marked as 🟡). 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.

Table of curves (see the lexicon below for explanation of the remarks)

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

Lexicon

M curve: Algebraic curve with the maximum number of connected components for its degree: $M = \frac{(d - 1) (d - 2)}{2} + 1$ .
M-k curve: Algebraic curve with $M - k$ connected components ( $k$ below the maximum).
EM-k curve: Extremal M-k curve: no extra oval can be added without changing the relative position of the existing ovals or increasing the degree.
LM-k curve: Extremal M-k curve: no extra oval can be added locally without changing the relative position of the existing ovals. Locally means that the transformation is a deformation in the space of polynomials with the same degree using only one singular point which is not an isolated point of the curve.
Harnacks's M-curve: Refers to the 1876 construction of Harnack [Harnack 1876] reaching the maximum for any degree.
Hilberts's M-curve: Refers to the 1891 construction of Hilbert [Hilbert 1891] reaching the maximum with nests containing many ovals. The construction also gives the two EM-2 sextic curves presented on this page.
Gudkov's M-curve: Refers to the 1960s construction of Gudkov [Gudkov 1974] for the last missing sextic M-curve.

Bibliography

A. Harnack
“Über die Vieltheiligkeit der ebenen algebraischen Curven”
Mathematische Annalen, 10 (1876), 189–199.
D. Hilbert
“Ueber die reellen Züge algebraischer Curven”
Mathematische Annalen, 38 (1891), 115–138.
D. A. Gudkov
“The topology of real projective algebraic varieties”
Russian Mathematical Surveys, 29 (1974), 3–79. (Earlier version exists in russian)

Exact polynomials

Degree 2, ⟨1⟩ (M curve) 🟢

\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 \pm 0 \end{array}

Degree 3, ⟨J ∐ 1⟩ (M curve) 🟢

\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 \pm 0 \end{array}

Degree 4, ⟨4⟩ (M curve) 🟢

\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 \pm 0 \end{array}

Degree 4, ⟨3⟩ (M-1 curve) 🟢

\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 \pm 0 \end{array}

Degree 6, ⟨10⟩ V1 (LM-1 curve) 🟢

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 \pm 0 \end{array}

Degree 6, ⟨10⟩ V2 (LM-1 curve) 🟢

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 \pm 0 \end{array}

Approximated polynomials

Degree 5, ⟨𝐽 ∐ 6⟩ (Harnack's M-curve) 🟢

\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) & = & 0.0024915028190241140454 \pm 2.1 \cdot 10^{-18} \end{array}

Degree 5, ⟨𝐽 ∐ 5⟩ (M-1 curve) 🟢

\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) & = & 0.023035530697453850865 \pm 2.0 \cdot 10^{-17} \end{array}

Degree 6, ⟨9 ∐ 1⟨1⟩⟩ (Harnack's M-curve) 🟢

\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) & = & 0.00009611423351077027294014 \pm 4.4 \cdot 10^{-17} \end{array}

Degree 6, ⟨6 ∐ 1⟨2⟩⟩ (EM-2 curve) 🟢

\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) & = & 0.0002488484400222630224711 \pm 1.4 \cdot 10^{-17} \end{array}

Degree 6, ⟨2 ∐ 1⟨6⟩⟩ (EM-2 curve) 🟡

\begin{array}{rcl} P & = & - 0.396663870421845 x^{6} + 0.21640136821548 x^{5} y - 0.0109419425863456 x^{5} z - 0.817302526479211 x^{4} y^{2} + 0.480727148033585 x^{4} y z + 1.27206642925114 x^{4} z^{2} + 0.0508999223953165 x^{3} y^{3} - 0.175349776021781 x^{3} y^{2} z - 0.451149966359067 x^{3} y z^{2} + 0.0232569761630186 x^{3} z^{3} - 0.163029808841434 x^{2} y^{4} + 1.04638338228826 x^{2} y^{3} z + 1.51195475956507 x^{2} y^{2} z^{2} - 1.04470250583701 x^{2} y z^{3} - 1.35783369451186 x^{2} z^{4} - 0.234906246896497 x y^{5} - 0.16662769621355 x y^{4} z + 0.0221136877114046 x y^{3} z^{2} + 0.181835573825952 x y^{2} z^{3} + 0.235225289182254 x y z^{4} - 0.0123739559359578 x z^{5} + 0.194998705706449 y^{6} + 0.566460762192212 y^{5} z - 0.0075389028994482 y^{4} z^{2} - 1.13007440315571 y^{3} z^{3} - 0.670730783712748 y^{2} z^{4} + 0.566435465154484 y z^{5} + 0.482446648620436 z^{6} \\ ≃ & - 17235.78104699570293473447 {(- x - 0.2851841792468901466164228 y - 0.9466525988899391577200791 z)}^{6} + 5899.186159204703864722619 {(- x - 0.1537138956698187087281564 y - 0.9312014362521081371866965 z)}^{6} + 11386.0057152624567782707 {(- x + 0.06854216572956906567952428 y + 0.9683731905951183589236783 z)}^{6} + 15282.44910132815415039056 {(- 0.9548768722027601019074167 x + 0.3723098942956923570212882 y + z)}^{6} + 8332.79310063802901769469 {(- 0.9524616191849832204727064 x - 0.6088550471512390858715661 y - z)}^{6} - 12283.77805358791429886764 {(- 0.7738609928751662362447604 x + 0.6246203299665835452853354 y + z)}^{6} - 7005.341876830503662653868 {(- 0.6889837205598905521157403 x - 0.8815560402753638775140902 y - z)}^{6} + 17902.62673388210327959808 {(- 0.6691231242757296513154339 x + 0.6996944209205973014384501 y - z)}^{6} + 10250.59896621212295068646 {(- 0.4826816020533194278651224 x + 0.854600981819704456623873 y + z)}^{6} - 8637.513759236634518120618 {(- 0.1074991467068560466435651 x + y + 0.9985571075818118589669307 z)}^{6} + 8228.001358636306327540664 {(0.3006376799025245188684201 x + y + 0.9806010843155804926650267 z)}^{6} - 1178.711579137469231076519 {(0.4249775146433635344137617 x - y + 0.9217325980973065315506236 z)}^{6} + 3450.569190891388066518074 {(0.5069223619506646982646841 x - 0.9772348022780100016635431 y + z)}^{6} - 9242.362451691050389456619 {(0.5827266931497678686261168 x - 0.8368117392224609922490641 y + z)}^{6} - 5161.022078622623169989155 {(0.6585249967682479015943296 x - 0.682249366830989123122004 y + z)}^{6} - 7176.85403354022042366776 {(0.771066115395020367890421 x - 0.5732827026747682488694338 y + z)}^{6} - 24264.08468939870918547297 {(x - 0.1699662094439660390551409 y - 0.9706045077988005008011373 z)}^{6} + 6651.727835992715249056793 {(x - 0.1176627272258781352437861 y - 0.9485086878956505224980934 z)}^{6} - 2784.698770358048095086016 {(x + 0.001988443429663307070758101 y - 0.9833693350705429068603123 z)}^{6} + 6426.084880811371644254234 {(x + 0.113795140982637392991505 y + 0.9509181896889256582375056 z)}^{6} \\ ‖ P ‖ & = & 1.0 \\ ‖ P - P_{1} ‖ & = & 1.563486337576989874767171 \cdot 10^{-11} \\ dist (P, Δ) & = & 0.0000006564983740133337636567442 \pm 2.2 \cdot 10^{-17} \end{array}

Degree 6, ⟨1 ∐ 1⟨9⟩⟩ (Hilbert's M curve) 🟢

\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) & = & 5.82146442783478375870245282 \cdot 10^{-9} \pm 2.6 \cdot 10^{-17} \end{array}

Degree 6, ⟨5 ∐ 1⟨5⟩⟩ (Gudkov's M curve) 🟢

\begin{array}{rcl} P & = & - 0.00132990000822424 y^{6} - 0.0616200461333061 y^{5} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) + 0.00104567643964428 y^{5} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) + 0.0423766780436503 y^{4} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) - 0.00223803203245656265329976974954 y^{4} {(x - z)}^{2} + 0.000244121531585367849579923382741 y^{4} {(x + z)}^{2} + 0.0285392824519227691903999755141 y^{3} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{2} - 0.12925458496204814129448834592 y^{3} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{2} + 0.000389754482938959525133847046016 y^{3} {(x - z)}^{3} + 0.000124842074475882852056131479617 y^{3} {(x + z)}^{3} + 0.0124919547764453464109096998163 y^{2} (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{3} - 0.00354416617249992576772138111049 y^{2} (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{3} + 0.00000242636930582478404659543119726 y^{2} {(x - z)}^{4} + 0.394230632890865184680961874619 y^{2} {(x - z)}^{2} {(x + z)}^{2} + 0.00000440611823967971157599027992591 y^{2} {(x + z)}^{4} + 0.000427153819177914190023670482077 y (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{4} + 0.000189092943319493398627445523985 y (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{4} - 0.000000247042452224817412433703003204 y {(x - z)}^{5} - 0.0165983876059155323147168928308 y {(x - z)}^{3} {(x + z)}^{2} + 0.172673791645410065331194659611 y {(x - z)}^{2} {(x + z)}^{3} - 0.0000000363005780728571624794169484397 y {(x + z)}^{5} - 0.00000166047194284043445679732716461 (0.707106781186547524400844362105 x - 0.707106781186547524400844362105 z) {(x + z)}^{5} + 0.0000225364436888175191844757124224 (0.707106781186547524400844362105 x + 0.707106781186547524400844362105 z) {(x - z)}^{5} - 0.0000000161812343280988403633336293434 {(x - z)}^{6} - 0.00515590206634466020202411939977 {(x - z)}^{4} {(x + z)}^{2} + 0.546067270146320171697595924343 {(x - z)}^{3} {(x + z)}^{3} + 0.00518676318548150563914056476733 {(x - z)}^{2} {(x + z)}^{4} + 6.84766574026006639940924253017 \cdot 10^{-11} {(x + z)}^{6} \\ ≃ & - 167462140.321176701583652489132 {(- x - 0.0841817828027667998013392961308 y + 0.997485287857214101376916696201 z)}^{6} + 55055739.9501972194156296177312 {(- x + 0.114364481107367551732842941469 y - 0.996212862730053857147131461678 z)}^{6} - 8280847.7360865321364889243944 {(- 0.981637388246154630186503873753 x - 0.608825284266300590089525078996 y - z)}^{6} - 70142245.4512914168820429031835 {(- 0.979707711316297963576049017957 x - 0.253739208032426041478352424778 y + z)}^{6} + 1862150.55171690285025281278593 {(- 0.97900135498655133776371225856 x - 0.696336260628075677058575824432 y - z)}^{6} + 38643974.7174452970042259593978 {(- 0.959486200702161240283936700576 x - 0.351627944323446397268378043049 y + z)}^{6} - 16838923.7252090821141439953013 {(- 0.937663269997385300680537406413 x - 0.437809661375089231717735299376 y + z)}^{6} + 4147402.57405932621001633301867 {(- 0.920482049980398491352772998286 x - 0.497097510591201133054486160271 y + z)}^{6} + 21347925.5652220917814409591214 {(0.985538655028765658353322541713 x + 0.48252183183334366208273910617 y + z)}^{6} - 43159828.9194993573114315403831 {(0.990012953172638185336852294883 x + 0.337758659905169118211280524351 y + z)}^{6} + 111921653.018760414725990086373 {(0.994346300208977238119674610775 x + 0.160049406947371112625592154293 y - z)}^{6} + 74322732.0203905853617009877828 {(0.99438795537665266811902468899 x + 0.193089178153952863624172832055 y + z)}^{6} - 113859450.916217918569333495353 {(0.998242726986911135550978449745 x + 0.0641542699349463333893876386532 y + z)}^{6} - 24090665.934844113584111080236 {(x - 0.146840460784293085682587852068 y + 0.992610038479574302394103475322 z)}^{6} + 97398696.9951930851082333271693 {(x - 0.127664630869451728516467679197 y + 0.994391918381327075311969256048 z)}^{6} - 223256453.985527753571557141228 {(x - 0.0988875479900851276194391345382 y + 0.996313881033065652189556837229 z)}^{6} + 162547806.184808119971082792091 {(x - 0.0361740380194784057704085566612 y + 0.998602810810778317617163788778 z)}^{6} + 26165484.2858941698483452614031 {(x - 0.00254821474123598904155948467865 y - 0.99464857619524777621020267677 z)}^{6} - 105380059.313954612248478721823 {(x + 0.0130316203394396028769926166129 y - 0.994109953477023694216423270649 z)}^{6} - 59617163.0160084518612508197032 {(x + 0.0242030510843510414704556666324 y - 0.993232004268181654090580090746 z)}^{6} + 238909354.144697984766328907855 {(x + 0.0358420066614339162037169348755 y - 0.994282639841816167146782195051 z)}^{6} \\ ‖ P ‖ & = & 1.00000000000000015344142921732 \\ ‖ P - P_{1} ‖ & = & 1.56879735402602020685295966417 \cdot 10^{-16} \\ dist (P, Δ) & = & 7.38323798474677818443979735078 \cdot 10^{-11} \pm 8.0 \cdot 10^{-21} \end{array}

Degree 7, ⟨𝐽 ∐ 15⟩ (Harnack's M curve) 🟢

\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) & = & 0.00000238287955329906566192218 \pm 2.8 \cdot 10^{-17} \end{array}

Degree 8, ⟨18 ∐ 1⟨3⟩⟩ (Harnack's M curve) 🟢

\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) & = & 3.4263980859463065386598227 \cdot 10^{-8} \pm 1.9 \cdot 10^{-17} \end{array}

Degree 8, ⟨19⟩ (M-3 curve) 🟢

This curve has 43 quasi-critical points, but the rank of the corresponding family of 8th power of linear forms is only 41, hence the polynomial is written using only 41 independent 8th power of linear forms. The missing points appear clearly on the drawing.

\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) & = & 0.00069817037013123745928 \pm 1.2 \cdot 10^{-17} \end{array}

Degree 9 ⟨J U 28⟩ (Harnack's M curve) 🔴

\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) & = & 2.4519823671465 \cdot 10^{-10} \pm 3.3 \cdot 10^{-14} \end{array}

Exceptional polynomials

Table of curves (see the lexicon below for explanation of the remarks)

Lexicon

Bibliography

Exact polynomials

Degree 2, ⟨1⟩ (M curve) 🟢Copy python definition

Degree 3, ⟨J ∐ 1⟩ (M curve) 🟢Copy python definition

Degree 4, ⟨4⟩ (M curve) 🟢Copy python definition

Degree 4, ⟨3⟩ (M-1 curve) 🟢Copy python definition

Degree 6, ⟨10⟩ V1 (LM-1 curve) 🟢Copy python definition

Degree 6, ⟨10⟩ V2 (LM-1 curve) 🟢Copy python definition

Approximated polynomials

Degree 5, ⟨𝐽 ∐ 6⟩ (Harnack's M-curve) 🟢Copy python definition

Degree 5, ⟨𝐽 ∐ 5⟩ (M-1 curve) 🟢Copy python definition

Degree 6, ⟨9 ∐ 1⟨1⟩⟩ (Harnack's M-curve) 🟢Copy python definition

Degree 6, ⟨6 ∐ 1⟨2⟩⟩ (EM-2 curve) 🟢Copy python definition

Degree 6, ⟨2 ∐ 1⟨6⟩⟩ (EM-2 curve) 🟡Copy python definition

Degree 6, ⟨1 ∐ 1⟨9⟩⟩ (Hilbert's M curve) 🟢Copy python definition

Degree 6, ⟨5 ∐ 1⟨5⟩⟩ (Gudkov's M curve) 🟢Copy python definition

Degree 7, ⟨𝐽 ∐ 15⟩ (Harnack's M curve) 🟢Copy python definition

Degree 8, ⟨18 ∐ 1⟨3⟩⟩ (Harnack's M curve) 🟢Copy python definition

Degree 8, ⟨19⟩ (M-3 curve) 🟢Copy python definition

Degree 9 ⟨J U 28⟩ (Harnack's M curve) 🔴Copy python definition