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\newcommand{\courseName}{Math 480A2}
\newcommand{\hwNumber}{5}
\newcommand{\hwDueDate}{September 29, 2022}


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\begin{document}

\begin{center}
	{\large \courseName, Homework \hwNumber}\\
	Due \hwDueDate\\
\end{center}

{\small \textit{Homework is graded out of a total of 10 points.  Collaboration is permitted, but you must list all coauthors on a problem's solution at the top of the page, and your writing must be your own.}}

\vspace{5mm}

\textbf{Problem 1.} (2 points) Let $E$ be the elliptic curve over $\RR$ defined by the short Weierstrass equation $y^2 = x^3 - x + 1$, and let $P = (1, 1)$, $Q = (-1, 1)$, and $R = (-1, -1)$ be points on $E$.  For both of the sums $P + Q$ and $P + R$, find the line $L$ connecting the summands, find the third intersection point of $L$ with $E$, and compute the value of the sum.

\vspace{3mm}

\textbf{Problem 2.} (2 points) Let $C$ be the curve over $\RR$ defined by the equation $f(x, y) = xy - 1 = 0$.  Give the homogenization of $f$, and find the points on the corresponding projective curve which lie on the line at infinity.  (\textit{Hint:} the points on the line at infinity correspond with nonzero solutions in $x, y, z$ satisfying $z=0$.)

\vspace{3mm}

\textbf{Problem 3.} (3 points) Let $C$ be the curve over $\RR$ defined by the equation $y = x^3 - x$.  For $i = 1, 2, 3$, find a line $L_i$ which intersects $C$ at a point with multiplicity $i$.

\vspace{3mm}

\textbf{Problem 4.} (3 points) Let $C$ be the curve over $\RR$ defined by the equation $y = x^3$.  The conclusion of B\'ezout's theorem states that a line (a curve defined by a polynomial of degree 1) should have three points of intersection with $C$ (a curve defined by a polynomial of degree 3), with several subtleties and caveats.  Each of the lines $y = 0$, $y = 1$, and $x = 0$ intersect $C$ at only a single point in $\RR^2$.  For each, explain the sense in which it intersects $C$ three times.

\end{document}