Quartic point group operation - Revision history

Rational Point: /* Solving for leading coefficient of resolvent */

2025-02-11T20:22:36Z

Solving for leading coefficient of resolvent

← Older revision		Revision as of 20:22, 11 February 2025
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	This equation is solved numerically in <big>'''[https://www.r-project.org/ R]'''</big>, and the first solution for ''a'' found by R’s <tt><span style="color:blue">polyroot</span>()</tt> function is verified to satisfy the resolvent quadratic equations and their derivatives, collectively known as the '''gunners’ equations''', with very small residuals both at the given point ''x''<sub>1</sub> and at the point ''x'' which is derived from ''a'' when the cubic equation is solved.		This equation is solved numerically in <big>'''[https://www.r-project.org/ R]'''</big>, and the first solution for ''a'' found by R’s <tt><span style="color:blue">polyroot</span>()</tt> function is verified to satisfy the resolvent quadratic equations and their derivatives, collectively known as the '''gunners’ equations''', with very small residuals both at the given point ''x''<sub>1</sub> and at the point ''x'' which is derived from ''a'' when the cubic equation is solved.

	<p style="overflow:auto;border:1pt solid;background-color:#8080ff">There is much unwritten history on these equations dating back to the Civil War in the United States. While the white Confederate belles were square-dancing with the gentlemen of the district, at formal balls catered by their black slaves, the Union railroad engineers and military officers were hard at work solving the gunners’ equations for the trajectories of cannonballs and artillery shells. And even at times of peace, nobody had ever heard of a square dance hall where gentlemanly duels and gunfights didn’t take place on a regular basis. Later on of course, Albert Einstein continued in the long tradition of gunners solving mathematical equations, to great and devastating effect on two particular Japanese small-town red-light disticts.</p>		<p style="overflow:auto;border:1pt solid;padding:1ex;background-color:#8080ff">There is much unwritten history on these equations dating back to the Civil War in the United States. While the white Confederate belles were square-dancing with the gentlemen of the district, at formal balls catered by their black slaves, the Union railroad engineers and military officers were hard at work solving the gunners’ equations for the trajectories of cannonballs and artillery shells. And even at times of peace, nobody had ever heard of a square dance hall where gentlemanly duels and gunfights didn’t take place on a regular basis. Later on of course, Albert Einstein continued in the long tradition of gunners solving mathematical equations, to great and devastating effect on two particular Japanese small-town red-light disticts.</p>

	The [[discriminant]] of the cubic equation for ''a'' turns out to be equal to zero, and the equation has both a single root and a double root; only the single root is useful for determining the point of tangency (''x'',''y'') we seek. By altering ''a'' slightly it must be possible to either overshoot or undershoot the solution slightly in the general case, and any double root must be cancelled out.		The [[discriminant]] of the cubic equation for ''a'' turns out to be equal to zero, and the equation has both a single root and a double root; only the single root is useful for determining the point of tangency (''x'',''y'') we seek. By altering ''a'' slightly it must be possible to either overshoot or undershoot the solution slightly in the general case, and any double root must be cancelled out.

Rational Point: /* The system of equations */ reword

2025-02-11T19:52:02Z

The system of equations: reword

← Older revision		Revision as of 19:52, 11 February 2025
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	\end{align}</math>		\end{align}</math>

			=== Solving for leading coefficient of resolvent ===
	This equation is solved numerically in <big>'''[https://www.r-project.org/ R]'''</big>, and the first solution for ''a'' found by R’s <tt><span style="color:blue">polyroot</span>()</tt> function is verified to satisfy the resolvent quadratic equations and their derivatives, collectively known as the '''gunners’ equations''', with very small residuals both at the given point ''x''<sub>1</sub> and at the point ''x'' which is derived from ''a'' when the cubic equation is solved.		This equation is solved numerically in <big>'''[https://www.r-project.org/ R]'''</big>, and the first solution for ''a'' found by R’s <tt><span style="color:blue">polyroot</span>()</tt> function is verified to satisfy the resolvent quadratic equations and their derivatives, collectively known as the '''gunners’ equations''', with very small residuals both at the given point ''x''<sub>1</sub> and at the point ''x'' which is derived from ''a'' when the cubic equation is solved.

	<p style="overflow:auto;border:1pt solid;background-color:#8080ff">There is much unwritten history on these equations dating back to the Civil War in the United States. While the white Confederate belles were square-dancing with the gentlemen of the district, at formal balls catered by their black slaves, the Union railroad engineers and military officers were hard at work solving the gunners’ equations for the trajectories of cannonballs and artillery shells. And even at times of peace, nobody had ever heard of a square dance hall where gentlemanly duels and gunfights didn’t take place on a regular basis. Later on of course, Albert Einstein continued in the long tradition of gunners solving mathematical equations, to great and devastating effect on two particular Japanese small-town red-light disticts.</p>		<p style="overflow:auto;border:1pt solid;background-color:#8080ff">There is much unwritten history on these equations dating back to the Civil War in the United States. While the white Confederate belles were square-dancing with the gentlemen of the district, at formal balls catered by their black slaves, the Union railroad engineers and military officers were hard at work solving the gunners’ equations for the trajectories of cannonballs and artillery shells. And even at times of peace, nobody had ever heard of a square dance hall where gentlemanly duels and gunfights didn’t take place on a regular basis. Later on of course, Albert Einstein continued in the long tradition of gunners solving mathematical equations, to great and devastating effect on two particular Japanese small-town red-light disticts.</p>

	The [[discriminant]] of the cubic equation for ''a'' turns out to be equal to zero, and the equation has both a single root and a double root; only the single root is useful for determining the point of tangency (''x'',''y'') we seek. By altering ''a'' slightly it must be possible to either overshoot or undershoot the solution slightly, and a double root must be cancelled out.		The [[discriminant]] of the cubic equation for ''a'' turns out to be equal to zero, and the equation has both a single root and a double root; only the single root is useful for determining the point of tangency (''x'',''y'') we seek. By altering ''a'' slightly it must be possible to either overshoot or undershoot the solution slightly in the general case, and any double root must be cancelled out.

	== Point averaging ==		== Point averaging ==

Rational Point: /* Additive inverses */ various corrections and additions

2025-02-11T19:48:24Z

Additive inverses: various corrections and additions

← Older revision		Revision as of 19:48, 11 February 2025
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	where the resolvent parabola <math>y=ax^2+bx+c</math> is required to be '''tangent''' <ref>Latin for “touching.”</ref> to the original curve both at the point ''P'' = (''x''<sub>1</sub>,''y''<sub>1</sub>) and at the unknown point ''Q'' = (''x'',''y'').		where the resolvent parabola <math>y=ax^2+bx+c</math> is required to be '''tangent''' <ref>Latin for “touching.”</ref> to the original curve both at the point ''P'' = (''x''<sub>1</sub>,''y''<sub>1</sub>) and at the unknown point ''Q'' = (''x'',''y'').

			=== Another long division ===
	We begin by dividing ''x'' – ''x''<sub>1</sub> into the quotient resulting from the previous division of the same binomial into the resultant polynomial.		We begin by dividing ''x'' – ''x''<sub>1</sub> into the quotient resulting from the previous division of the same binomial into the resultant polynomial.

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	&&&{b^2+2ac-q_2+4abx_1\choose-2q_3x_1+3a^2x_1^2-3q_4x_1^2}x&{}+{2bc-q_1+b^2x_1+2acx_1-q_2x_1\choose+ 2abx_1^2-q_3x_1^2+a^2x_1^3-q_4x_1^3}\\		&&&{b^2+2ac-q_2+4abx_1\choose-2q_3x_1+3a^2x_1^2-3q_4x_1^2}x&{}+{2bc-q_1+b^2x_1+2acx_1-q_2x_1\choose+ 2abx_1^2-q_3x_1^2+a^2x_1^3-q_4x_1^3}\\
	&&&{b^2+2ac-q_2+4abx_1\choose-2q_3x_1+3a^2x_1^2-3q_4x_1^2}x&{}-{b^2x_1+2acx_1-q_2x_1+4abx_1^2\choose-2q_3x_1^2+3a^2x_1^3-3q_4x_1^3}\\\hline		&&&{b^2+2ac-q_2+4abx_1\choose-2q_3x_1+3a^2x_1^2-3q_4x_1^2}x&{}-{b^2x_1+2acx_1-q_2x_1+4abx_1^2\choose-2q_3x_1^2+3a^2x_1^3-3q_4x_1^3}\\\hline
	R=&&&&{2bc-q_1+2b^2x_1+4acx_1-2q_2x_1\choose +6abx_1^2-3q_3x_1^2+3a^2x_1^3-~~3q_4x_1~~^3}		R=&&&&{2bc-q_1+2b^2x_1+4acx_1-2q_2x_1\choose +6abx_1^2-3q_3x_1^2+4a^2x_1^3-4q_4x_1^3}
	\end{array}</math>		\end{array}</math>
	[[Image:Quartic point inverse.svg\|frame\|right\|Graphical depiction of inverse points on a quartic curve, showing the two points of tangency of the parabola with the quartic curve, and the vertical line which bisects the line segment between the inverse points and also intersects at the same point where the tangent lines do.]]		[[Image:Quartic point inverse.svg\|frame\|right\|Graphical depiction of inverse points on a quartic curve, showing the two points of tangency of the parabola with the quartic curve, and the vertical line which bisects the line segment between the inverse points and also intersects at the same point where the tangent lines do.]]

	This remainder, on dividing ''x'' – ''x''<sub>1</sub> into the resultant polynomial for the second time, is equal to the ''derivative'' of the original resultant at the point ''x''<sub>1</sub>, and may be set equal to zero. We are now looking for the second point of tangency ''x'' between the quartic curve and the parabola of the resolvent quadratic equation. Hence four equations to be solved simultaneously for ''a, b, c'' and ''x'' in terms of ''q''<sub>4</sub>, ''q''<sub>3</sub>, ''q''<sub>2</sub>, ''q''<sub>1</sub>, ''q''<sub>0</sub>, ''x''<sub>1</sub>, ''y''<sub>1</sub> and ''y''<sub>1</sub>’.		This remainder, on dividing ''x'' – ''x''<sub>1</sub> into the resultant polynomial for the second time, is equal to the ''derivative'' of the original resultant at the point ''x''<sub>1</sub>, and may be set equal to zero as part of system of equations; there are many different ways to express the system of equations we need to solve, and this may not be the simplest.

			=== The system of equations ===

			In any case we are now looking for the second point of tangency ''x'' between the quartic curve and the parabola of the resolvent quadratic equation. Hence four equations to be solved simultaneously for ''a, b, c'' and ''x'' in terms of ''q''<sub>4</sub>, ''q''<sub>3</sub>, ''q''<sub>2</sub>, ''q''<sub>1</sub>, ''q''<sub>0</sub>, ''x''<sub>1</sub>, ''y''<sub>1</sub> and ''y''<sub>1</sub>’ may be derived most simply by passing the parabola through the point (''x''<sub>1</sub>,''y''<sub>1</sub>) with a slope or derivative of ''y''<sub>1</sub>’ at that point, and then setting both the quotient from the last division and its derivative equal to zero at the point (''x'',''y'').


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	This equation is solved numerically in <big>'''[https://www.r-project.org/ R]'''</big>, and the first solution for ''a'' found by R’s <tt><span style="color:blue">polyroot</span>()</tt> function is verified to satisfy the resolvent quadratic equations and their derivatives, collectively known as the '''gunners’ equations''', with very small residuals both at the given point ''x''<sub>1</sub> and at the point ''x'' which is derived from ''a'' when the cubic equation is solved.		This equation is solved numerically in <big>'''[https://www.r-project.org/ R]'''</big>, and the first solution for ''a'' found by R’s <tt><span style="color:blue">polyroot</span>()</tt> function is verified to satisfy the resolvent quadratic equations and their derivatives, collectively known as the '''gunners’ equations''', with very small residuals both at the given point ''x''<sub>1</sub> and at the point ''x'' which is derived from ''a'' when the cubic equation is solved.

	There is much unwritten history on these equations dating back to the Civil War in the United States. While the white Confederate belles were square-dancing with the gentlemen of the district, at formal balls catered by their black slaves, the Union railroad engineers and military officers were hard at work solving the gunners’ equations for the trajectories of cannonballs and artillery shells. And even at times of peace, nobody had ever heard of a square dance hall where gentlemanly duels and gunfights didn’t take place on a regular basis. Later on of course, Albert Einstein continued in the long tradition of gunners solving mathematical equations, to great and devastating effect on two particular Japanese small-town red-light disticts.		<p style="overflow:auto;border:1pt solid;background-color:#8080ff">There is much unwritten history on these equations dating back to the Civil War in the United States. While the white Confederate belles were square-dancing with the gentlemen of the district, at formal balls catered by their black slaves, the Union railroad engineers and military officers were hard at work solving the gunners’ equations for the trajectories of cannonballs and artillery shells. And even at times of peace, nobody had ever heard of a square dance hall where gentlemanly duels and gunfights didn’t take place on a regular basis. Later on of course, Albert Einstein continued in the long tradition of gunners solving mathematical equations, to great and devastating effect on two particular Japanese small-town red-light disticts.</p>

	~~A simpler symbolic~~ solution ~~still needs to~~ be ~~worked~~ out ~~for rational points on quartic curves~~.		The [[discriminant]] of the cubic equation for ''a'' turns out to be equal to zero, and the equation has both a single root and a double root; only the single root is useful for determining the point of tangency (''x'',''y'') we seek. By altering ''a'' slightly it must be possible to either overshoot or undershoot the solution slightly, and a double root must be cancelled out.

	== Point averaging ==		== Point averaging ==

Rational Point: biquadratic

2025-02-07T19:31:15Z

biquadratic

← Older revision		Revision as of 19:31, 7 February 2025
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	[[Image:Quart.svg\|frame\|right\|A “twisted infinity-bun” curve of degree four, which may or may not have many rational points or be suitable for cryptographic use.]]		[[Image:Quart.svg\|frame\|right\|A “twisted infinity-bun” curve of degree four, which may or may not have many rational points or be suitable for cryptographic use.]]
	The [[point group operation]] over quartic (or hyperelliptic <ref>https://hyperelliptic.org/</ref>~~) curves (~~of degree four) is defined geometrically as follows. Let		The [[point group operation]] over a '''quartic''' (or, rarely, '''biquadratic''') '''curve''' (i.e., a '''hyperelliptic''' <ref>https://hyperelliptic.org/</ref> curve of degree four) in ''y''<sup>2</sup> is defined geometrically as follows. Let
	:<math>y^2 = q_4x^4+q_3x^3+q_2x^2+q_1x+q_0</math>		:<math>y^2 = q_4x^4+q_3x^3+q_2x^2+q_1x+q_0</math>
	be a quartic curve on the ''x''-''y'' plane of real numbers, the polynomial in ''x'' on the right having either the coefficient ''q''<sub>4</sub>>0 or at least two real roots counting multiplicity. Quartic curves of this form include various lemniscates and ovals with a negative leading coefficient as well as open curves with a positive leading coefficient and two or three disconnected components.		be a quartic curve on the ''x''-''y'' plane of real numbers, the polynomial in ''x'' on the right having either the coefficient ''q''<sub>4</sub>>0 or at least two real roots counting multiplicity. Quartic curves of this form include various lemniscates and ovals with a negative leading coefficient as well as open curves with a positive leading coefficient and two or three disconnected components.

Rational Point: /* A quartic square dance */

2025-02-07T17:07:59Z

A quartic square dance

← Older revision		Revision as of 17:07, 7 February 2025
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	Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.		Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.

	More general quartic curves in the plane might be met with more general conics having parallel axes to serve as resolvents, unless they are first brought into a canonical form in ''y''<sup>2</sup> by a conformal mapping, but be warned that the “group law” introduced by Bernstein, Edwards, Hamburg et al. for government-approved “elliptic” curve cryptography schemes [[Ed25519]] and [[Ed448]] promulgated in in official government standards such as [https://www.rfc-editor.org/rfc/rfc7748 RFC 7748], is essentially of the second degree only in character, and does not satisfy the proper resolvent equations for quartic curves which are not simply conics with squared coördinates.		More general quartic curves in the plane might be met with more general conics having parallel axes to serve as resolvents, unless they are first brought into a canonical form in ''y''<sup>2</sup> by a conformal mapping, but be warned that the “group law” introduced by Bernstein, Edwards, Hamburg et al. for government-approved “elliptic” curve cryptography schemes [[Ed25519]] and [[Ed448]] promulgated <ref>There is language that the government does not wish to be inferred to say that it has '''“promulgated”''' various technical schemes and protocols as official internet standards, that is to say it published the schemes on a less formal “Request-For-Comment” or hats-off-for-the-ladies “RFC” basis only. The government with that language is attempting to slam the brakes on the train and stop short of a legal position of having '''“homologated”''' certain proposed cryptographic schemes and not others for use by the general public, as particularly in Europe that tends to imply an undue, unlawful or unconstitutional arbitrary rule-making authority on the part of undirected and meddlesome government officials in technical matters.</ref> in in official government standards such as [https://www.rfc-editor.org/rfc/rfc7748 RFC 7748], is essentially of the second degree only in character, and does not satisfy the proper resolvent equations for quartic curves which are not simply conics with squared coördinates.

	== Additive inverses ==		== Additive inverses ==

Rational Point: be more descriptive

2025-02-07T08:40:21Z

be more descriptive

← Older revision		Revision as of 08:40, 7 February 2025
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	Every operation over a quartic curve will be defined geometrically by a “square dance” of the four points of intersection of the quartic curve and a parabola <ref>Borrowed perhaps from [[:Category:conic section cryptography]] but that is another matter.</ref> whose axis is parallel to the ''y''-axis, and otherwise to be determined.		Every operation over a quartic curve will be defined geometrically by a “square dance” of the four points of intersection of the quartic curve and a parabola <ref>Borrowed perhaps from [[:Category:conic section cryptography]] but that is another matter.</ref> whose axis is parallel to the ''y''-axis, and otherwise to be determined.

	~~Namely if~~ ''P, Q, R,'' and ''S'' are four points of the quartic curve, through which a parabola with an axis parallel to the ''y''-axis passes, then we will say that		If ''P, Q, R,'' and ''S'' are four points of the quartic curve, through which a parabola with an axis parallel to the ''y''-axis passes, then we will say that

	:<math>P\oplus Q\oplus R\oplus S=O</math>		:<math>P\oplus Q\oplus R\oplus S=O</math>
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	Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.		Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.

	More general quartic curves in the plane might be met with more general conics having parallel axes to serve as resolvents, unless they are first brought into a canonical form in ''y''<sup>2</sup> by a conformal mapping, but be warned that the “group law” introduced by Bernstein, Edwards, Hamburg et al. for government-approved “elliptic” curve cryptography schemes [[Ed25519]] and [[Ed448]] promulgated in ~~the~~ [https://www.rfc-editor.org/rfc/rfc7748 RFC 7748] ~~government standard~~, is essentially of the second degree only in character, and does not satisfy the proper resolvent equations for quartic curves which are not simply conics with squared coördinates.		More general quartic curves in the plane might be met with more general conics having parallel axes to serve as resolvents, unless they are first brought into a canonical form in ''y''<sup>2</sup> by a conformal mapping, but be warned that the “group law” introduced by Bernstein, Edwards, Hamburg et al. for government-approved “elliptic” curve cryptography schemes [[Ed25519]] and [[Ed448]] promulgated in in official government standards such as [https://www.rfc-editor.org/rfc/rfc7748 RFC 7748], is essentially of the second degree only in character, and does not satisfy the proper resolvent equations for quartic curves which are not simply conics with squared coördinates.

	== Additive inverses ==		== Additive inverses ==
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	This equation is solved numerically in <big>'''[https://www.r-project.org/ R]'''</big>, and the first solution for ''a'' found by R’s <tt><span style="color:blue">polyroot</span>()</tt> function is verified to satisfy the resolvent quadratic equations and their derivatives, collectively known as the '''gunners’ equations''', with very small residuals both at the given point ''x''<sub>1</sub> and at the point ''x'' which is derived from ''a'' when the cubic equation is solved.		This equation is solved numerically in <big>'''[https://www.r-project.org/ R]'''</big>, and the first solution for ''a'' found by R’s <tt><span style="color:blue">polyroot</span>()</tt> function is verified to satisfy the resolvent quadratic equations and their derivatives, collectively known as the '''gunners’ equations''', with very small residuals both at the given point ''x''<sub>1</sub> and at the point ''x'' which is derived from ''a'' when the cubic equation is solved.

	There is much unwritten history on these equations dating back to the Civil War in the United States. While the white Confederate belles were square-dancing with the gentlemen of the district, at balls catered by their black slaves, the Union railroad engineers and military officers were hard at work solving the gunners’ equations for the trajectories of cannonballs and artillery shells. And even at times of peace, nobody had ever heard of a square dance hall where gentlemanly duels and gunfights didn’t take place on a regular basis. Later on of course, Albert Einstein continued in the long tradition of gunners solving mathematical equations, to great and devastating effect on two particular Japanese small-town red-light disticts.		There is much unwritten history on these equations dating back to the Civil War in the United States. While the white Confederate belles were square-dancing with the gentlemen of the district, at formal balls catered by their black slaves, the Union railroad engineers and military officers were hard at work solving the gunners’ equations for the trajectories of cannonballs and artillery shells. And even at times of peace, nobody had ever heard of a square dance hall where gentlemanly duels and gunfights didn’t take place on a regular basis. Later on of course, Albert Einstein continued in the long tradition of gunners solving mathematical equations, to great and devastating effect on two particular Japanese small-town red-light disticts.

	A simpler symbolic solution still needs to be worked out for rational points on quartic curves.		A simpler symbolic solution still needs to be worked out for rational points on quartic curves.

Rational Point: /* A quartic square dance */

2025-02-05T19:44:44Z

A quartic square dance

← Older revision		Revision as of 19:44, 5 February 2025
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	Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.		Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.

	More general quartic curves in the plane might be met with more general conics having parallel axes to serve as resolvents, unless they are first brought into a canonical form in ''y''<sup>2</sup> by a conformal mapping, but be warned that the “group law” introduced by Bernstein, Edwards, Hamburg et al. for government-approved “elliptic” curve cryptography schemes [[Ed25519]] and [[Ed448]] ~~and~~ promulgated in the [https://www.rfc-editor.org/rfc/rfc7748 RFC 7748] government standard, is essentially of the second degree only in character, and does not satisfy the proper resolvent equations for quartic curves which are not simply conics with squared coördinates.		More general quartic curves in the plane might be met with more general conics having parallel axes to serve as resolvents, unless they are first brought into a canonical form in ''y''<sup>2</sup> by a conformal mapping, but be warned that the “group law” introduced by Bernstein, Edwards, Hamburg et al. for government-approved “elliptic” curve cryptography schemes [[Ed25519]] and [[Ed448]] promulgated in the [https://www.rfc-editor.org/rfc/rfc7748 RFC 7748] government standard, is essentially of the second degree only in character, and does not satisfy the proper resolvent equations for quartic curves which are not simply conics with squared coördinates.

	== Additive inverses ==		== Additive inverses ==

Rational Point: /* A quartic square dance */ “group law”

2025-02-05T19:20:29Z

A quartic square dance: “group law”

← Older revision		Revision as of 19:20, 5 February 2025
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	Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.		Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.

			More general quartic curves in the plane might be met with more general conics having parallel axes to serve as resolvents, unless they are first brought into a canonical form in ''y''<sup>2</sup> by a conformal mapping, but be warned that the “group law” introduced by Bernstein, Edwards, Hamburg et al. for government-approved “elliptic” curve cryptography schemes [[Ed25519]] and [[Ed448]] and promulgated in the [https://www.rfc-editor.org/rfc/rfc7748 RFC 7748] government standard, is essentially of the second degree only in character, and does not satisfy the proper resolvent equations for quartic curves which are not simply conics with squared coördinates.

	== Additive inverses ==		== Additive inverses ==

Rational Point: /* A quartic square dance */ gunners’ equations

2025-02-05T17:12:45Z

A quartic square dance: gunners’ equations

← Older revision		Revision as of 17:12, 5 February 2025
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	The equation of this parabola is known as the '''resolvent equation''' of the point group operation, and a '''resultant equation''' is determined whose roots are the four points of intersection of the resolvent parabolic trajectory with the quartic curve, called the '''solvend''' as it is the main equation to be solved, in ''y''<sup>2</sup>, for rational points which satisfy the point group operation.		The equation of this parabola is known as the '''resolvent equation''' of the point group operation, and a '''resultant equation''' is determined whose roots are the four points of intersection of the resolvent parabolic trajectory with the quartic curve, called the '''solvend''' as it is the main equation to be solved, in ''y''<sup>2</sup>, for rational points which satisfy the point group operation.

	Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, but still essential.		Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, as they inevitably involve derivatives, and are actually systems of differential equations, but are still essential to performing the point group operation over a quartic curve.

	== Additive inverses ==		== Additive inverses ==

Rational Point: /* A quartic square dance */

2025-02-05T17:04:45Z

A quartic square dance

← Older revision		Revision as of 17:04, 5 February 2025
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	The equation of this parabola is known as the '''resolvent equation''' of the point group operation, and a '''resultant equation''' is determined whose roots are the four points of intersection of the resolvent parabolic trajectory with the quartic curve, called the '''solvend''' as it is the main equation to be solved, in ''y''<sup>2</sup>, for rational points which satisfy the point group operation.		The equation of this parabola is known as the '''resolvent equation''' of the point group operation, and a '''resultant equation''' is determined whose roots are the four points of intersection of the resolvent parabolic trajectory with the quartic curve, called the '''solvend''' as it is the main equation to be solved, in ''y''<sup>2</sup>, for rational points which satisfy the point group operation.

	Equations involving four ~~distict~~ points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, but still essential.		Equations involving four distinct points are easiest, since any points of tangency require solving the '''gunners’ equations''' to fit parabolic trajectories, and these are much more difficult, but still essential.

	== Additive inverses ==		== Additive inverses ==