Weierstraß normal form: Difference between revisions
Weierstraß normal form - homework problem ?? |
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== General equation == | == General equation == | ||
[[Image:Karl-weierstrass-1-39ba31-640.jpg|right|frame|Karl | [[Image:Karl-weierstrass-1-39ba31-640.jpg|right|frame|Karl | ||
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== Problem == | == Problem == | ||
Substitute <math>\alpha x + \beta y + \zeta</math> and <math>\gamma x + \delta y + \eta</math> for ''z'' and ''w'' in the general equation, simplify by collecting like terms in respective powers of ''x'' and ''y'', and solve for ''α, β, γ, δ, ζ, η, a'' and ''b'' so that | Substitute <math>\alpha x + \beta y + \zeta</math> and <math>\gamma x + \delta y + \eta</math> for ''z'' and ''w'' in the general equation, simplify by collecting like terms in respective powers of ''x'' and ''y'', and solve for ''α, β, γ, δ, ζ, η, a'' and ''b'' so that<ref>Arnold Kas. “Weierstrass Normal Forms and Invariants of Elliptic Surfaces.” ''Transactions of the American Mathematical Society,'' vol. 225, Jan 1977, pp. 259-266. [https://www.ams.org/journals/tran/1977-225-00/S0002-9947-1977-0422285-X/S0002-9947-1977-0422285-X.pdf PDF]</ref> | ||
:<math>y^2 = x^3 + ax + b.</math> | :<math>y^2 = x^3 + ax + b.</math> | ||
Revision as of 22:30, 19 December 2024
General equation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gz^3 + hz^2w + jzw^2 + kw^3}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {} + rz + sw}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {} + t \qquad = 0.}
Linear transformation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} z \\ w \end{bmatrix} = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = \alpha x + \beta y + \zeta}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w = \gamma x + \delta y + \eta}
Problem
Substitute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha x + \beta y + \zeta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma x + \delta y + \eta} for z and w in the general equation, simplify by collecting like terms in respective powers of x and y, and solve for α, β, γ, δ, ζ, η, a and b so that[1]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 = x^3 + ax + b.}
