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NT2009HW8 - 4400/6400 PROBLEM SET 7 Recommendation 4400...

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4400/6400 PROBLEM SET 7 Recommendation: 4400 students should do at least 4 problems; 6400 students should do at least 6. 7.1) Write out a careful proof of Lemma 6 in the Minkowski’s Theorem handout. Especially, say a bit about the change-of-volume properties of the determinant map. Citing a reference or two might be appropriate. 7.2)** Prove Theorem 11 in the Minkowski’s theorem handout. (Possible strat- egy: talk to Professor Joseph H.G. Fu.) 7.3) Let Ω R N be a convex body with volume exactly 2 N . (We have seen that Ω need not have a nonzero lattice point.) Show that if Ω is moreover closed , then it does have a nonzero lattice point. 1 Suggestion: Use the fact that for all > 0, the dilate (1 + )Ω must contain a nonzero lattice point. 7.4)* Prove Theorem 12 in the Minkowksi’s Theorem handout. 7.5) Prove Lemma 14 (Euler’s identity) in the Minkowksi’s Theorem handout. 7.6) Prove Lemma 16 in the Minkowksi’s Theorem handout. 7.7) The three squares theorem of Legendre-Gauss says that a positive inte- ger n
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