Height - Revision history

Rational Point: logarithm base (basic math)

2025-02-06T21:24:17Z

logarithm base (basic math)

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	Height is used in the method of infinite descent to prove that some property is true of all rational numbers, when it can be shown that the property is true of any rational number whenever it is true of all rational numbers of lesser height.		Height is used in the method of infinite descent to prove that some property is true of all rational numbers, when it can be shown that the property is true of any rational number whenever it is true of all rational numbers of lesser height.

	The '''logarithmic height''' of ~~the same~~ rational number is then defined		The '''logarithmic height''' of a rational number is then defined as the logarithm <ref>Assuming a natural logarithm, but that is normally left implicit, and the reader should work out any consequences of choosing a base for the logarithm.</ref> of its rational height.

	:<math>h\left(\frac m n\right) = \log H\left(\frac m n\right)</math>.		:<math>h\left(\frac m n\right) = \log H\left(\frac m n\right)</math>.

Rational Point: /* Example program */ dependencies

2025-02-06T21:18:19Z

Example program: dependencies

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	with an ''x''-coördinate whose height does not exceed 1000.		with an ''x''-coördinate whose height does not exceed 1000.

	C++. Requires flags “<tt>-lgmpxx -lgmp</tt>” passed to compiler to link.		C++. Requires [https://gmplib.org/ GNU Multiple Precision “Bignum” Arithmetic Library] to be installed and flags “<tt>-lgmpxx -lgmp</tt>” passed to the compiler to link.
	<syntaxhighlight lang="C++" line="1">		<syntaxhighlight lang="C++" line="1">
	#include <iostream>		#include <iostream>

Rational Point: logarithmic and canonical heights

2025-02-06T21:01:25Z

logarithmic and canonical heights

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	The '''height''' <ref>Stephen Hoel Schanuel. “Heights in number fields.” ''Bulletin de la S. M. F.'', vol. 107 (1979), p. 433-449. http://www.numdam.org/item/?id=BSMF_1979__107__433_0</ref> of a rational number is defined to be the greater of the absolute values of its numerator and denominator in lowest terms.		The '''height''' <ref>Stephen Hoel Schanuel. “Heights in number fields.” ''Bulletin de la S. M. F.'', vol. 107 (1979), p. 433-449. http://www.numdam.org/item/?id=BSMF_1979__107__433_0</ref> (or '''rational height''') of a rational number is defined to be the greater of the absolute values of its numerator and denominator in lowest terms.

	If <math>\gcd(m,n)=1</math>, then <math>H\left(\frac m n\right) = \max(\|m\|,\|n\|)</math>.		If <math>\gcd(m,n)=1</math>, then <math>H\left(\frac m n\right) = \max(\|m\|,\|n\|)</math>.

	Height is used in the method of infinite descent to prove that some property is true of all rational numbers, when it can be shown that the property is true of any rational number whenever it is true of all rational numbers of lesser height.		Height is used in the method of infinite descent to prove that some property is true of all rational numbers, when it can be shown that the property is true of any rational number whenever it is true of all rational numbers of lesser height.

			The '''logarithmic height''' of the same rational number is then defined

			:<math>h\left(\frac m n\right) = \log H\left(\frac m n\right)</math>.

			The height of any rational point ''P'' on an elliptic curve ''E''/ℚ, when ''E'' is in [[Weierstraß normal form]], is typically understood to be the height of its ''x''-coördinate. The Néron–Tate '''canonical height''' of an elliptic curve ''E''/ℚ, starting at a rational point ''P'' is then defined

			:<math>\hat h_{E/\mathbb Q}(P)=\lim_{N\rightarrow\infty}
			\frac{h\left(\underbrace{P\oplus P\oplus \cdots\oplus P}_{N\times}\right)}{N^2}</math>,

			by iterating its [[point group operation]] ⊕ and taking a limit. Namely, Néron and Tate had observed that “doubling” a rational point on an elliptic curve typically raises its rational height to the fourth power in an asymptotic or limiting sense.

	== Example program ==		== Example program ==

Rational Point: /* Example program */ Not an issue in any of these examples, but, forgot ... Y2.canonicalize();

2025-02-05T20:56:05Z

Example program: Not an issue in any of these examples, but, forgot ... Y2.canonicalize();

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	X.get_num() = N; X.get_den() = D; X.canonicalize();		X.get_num() = N; X.get_den() = D; X.canonicalize();
	if (X.get_den() < D) continue; // not in lowest terms		if (X.get_den() < D) continue; // not in lowest terms
	Y2 = mpq_class(-7,17)X(X - 5)(X + 5)(X + 1);		Y2 = mpq_class(-7,17)X(X - 5)(X + 5)(X + 1); Y2.canonicalize();
	if (Y2 < 0) continue;		if (Y2 < 0) continue;
	n = sqrt(Y2.get_num()); Y.get_num() = n;		n = sqrt(Y2.get_num()); Y.get_num() = n;

Rational Point: /* Example program */ spacing

2025-02-05T06:46:51Z

Example program: spacing

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	A third curve, <math>y^2=-(x - 4)(x + 3)(x^2 + 1)</math>, does not seem to have any rational points of accessible height at all except for <math>(-3,0)</math>, <math>\left(-\frac{1352}{733}, \pm\frac{2929542}{537289}\right)</math>, <math>\left(-\frac12,\pm\frac{15}{4}\right)</math>, <math>\left(\frac35,\pm\frac{102}{25}\right)</math>, <math>\left(\frac{1077}{481},\pm\frac{1723260}{231361}\right)</math>, and <math>(4,0)</math>.		A third curve, <math>y^2=-(x - 4)(x + 3)(x^2 + 1)</math>, does not seem to have any rational points of accessible height at all except for <math>(-3,0)</math>, <math>\left(-\frac{1352}{733}, \pm\frac{2929542}{537289}\right)</math>, <math>\left(-\frac12,\pm\frac{15}{4}\right)</math>, <math>\left(\frac35,\pm\frac{102}{25}\right)</math>, <math>\left(\frac{1077}{481},\pm\frac{1723260}{231361}\right)</math>, and <math>(4,0)</math>.

	[[File:Rational points height 3000.svg]]		[[File:Rational points height 3000.svg]]

Rational Point: /* Example program */ math formula

2025-02-05T06:46:04Z

Example program: math formula

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	This example outputs all rational points on the curve		This example outputs all rational points on the curve

	:<math>y^2 = \frac{-7}{17}x(x-5)(x+5)(x+1)</math>		:<math>y^2 = -\frac{7}{17}x(x-5)(x+5)(x+1)</math>

	with an ''x''-coördinate whose height does not exceed 1000.		with an ''x''-coördinate whose height does not exceed 1000.

Rational Point: very few rational points

2025-02-05T06:45:18Z

very few rational points

← Older revision		Revision as of 06:45, 5 February 2025
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	A third curve, <math>y^2=-(x - 4)(x + 3)(x^2 + 1)</math>, does not seem to have any rational points of accessible height at all except for <math>(-3,0)</math>, <math>\left(-\frac{1352}{733}, \pm\frac{2929542}{537289}\right)</math>, <math>\left(-\frac12,\pm\frac{15}{4}\right)</math>, <math>\left(\frac35,\pm\frac{102}{25}\right)</math>, <math>\left(\frac{1077}{481},\pm\frac{1723260}{231361}\right)</math>, and <math>(4,0)</math>.		A third curve, <math>y^2=-(x - 4)(x + 3)(x^2 + 1)</math>, does not seem to have any rational points of accessible height at all except for <math>(-3,0)</math>, <math>\left(-\frac{1352}{733}, \pm\frac{2929542}{537289}\right)</math>, <math>\left(-\frac12,\pm\frac{15}{4}\right)</math>, <math>\left(\frac35,\pm\frac{102}{25}\right)</math>, <math>\left(\frac{1077}{481},\pm\frac{1723260}{231361}\right)</math>, and <math>(4,0)</math>.
			[[File:Rational points height 3000.svg]]

Rational Point: third curve

2025-02-05T06:32:14Z

third curve

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	[[File:Rational points height 100.svg]]		[[File:Rational points height 100.svg]]

			A third curve, <math>y^2=-(x - 4)(x + 3)(x^2 + 1)</math>, does not seem to have any rational points of accessible height at all except for <math>(-3,0)</math>, <math>\left(-\frac{1352}{733}, \pm\frac{2929542}{537289}\right)</math>, <math>\left(-\frac12,\pm\frac{15}{4}\right)</math>, <math>\left(\frac35,\pm\frac{102}{25}\right)</math>, <math>\left(\frac{1077}{481},\pm\frac{1723260}{231361}\right)</math>, and <math>(4,0)</math>.

Rational Point: /* Possible conclusions and further questions */

2025-02-05T05:27:18Z

Possible conclusions and further questions

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	* Is it too simple?		* Is it too simple?
	* What is the algebraic [[genus]] of the curve, and how does that affect the cryptographic group operations to be performed on it?		* What is the algebraic [[genus]] of the curve, and how does that affect the cryptographic group operations to be performed on it?

			Another curve with a slightly different equation has many, many more rational points.

			:<math>y^2 = -\frac7{13}(x-5)(x+5)\left(x^2+x+\frac14\right)</math>

			[[File:Rational points height 100.svg]]

Rational Point: output data plotted

2025-02-04T23:43:13Z

output data plotted

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	}		}
	</syntaxhighlight>		</syntaxhighlight>
			[[Image:Rational points height 1000.svg\|frame\|right\|Output data plotted in R.]]
	<pre>		<pre>
	-1, 0		-1, 0