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{\large \courseName, Homework \hwNumber}\\
Due \hwDueDate\\
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{\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.}}
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\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.
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\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 $z=0$.)
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\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 $i$.
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\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.
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