By James D. Lewis

ISBN-10: 0821805681

ISBN-13: 9780821805688

This booklet offers an advent to a subject of significant curiosity in transcendental algebraic geometry: the Hodge conjecture. which includes 15 lectures plus addenda and appendices, the quantity is predicated on a chain of lectures added by means of Professor Lewis on the Centre de Recherches Mathematiques (CRM). The ebook is a self-contained presentation, thoroughly dedicated to the Hodge conjecture and similar themes. It contains many examples, and such a lot effects are thoroughly confirmed or sketched. the incentive in the back of some of the effects and historical past fabric is equipped. This entire method of the booklet offers it a ``user-friendly'' type. Readers needn't seek in other places for varied effects. The ebook is acceptable to be used as a textual content for a issues path in algebraic geometry; comprises an appendix by means of B. Brent Gordon.

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What might be surprising at first sight is that there does not always exist a Hadamard matrix—for some n, they exist, and for other n, they do not. For example, there is a 2 × 2 Hadamard matrix but not a 3 × 3 one. What is perhaps even more surprising is that, in spite of the fact that the above question arose from an analytic perspective, Hadamard matrices are related more to coding theory, number theory, and combinatorics [vLW], [Ho]! In fact, a linear code constructed from a Hadamard matrix was used in the 1971 Mariner 9 mission to Mars.

6 Quadratic Residue Codes and Other Group Codes 25 on the additive group of the field of real numbers R. ) First, for the finite-field analog of eixy , we need to know how to construct the additive characters of F. Recall that an additive character of F is a function ψ : F → C for which ψ(x + y) = ψ(x)ψ(y). Let p > 2 denote an odd prime, and let ( pa ) denote the Legendre character: ⎧ ⎪ a = 0 quadratic residue mod p, ⎨ 1, a = −1, a = 0 quadratic nonresidue mod p, ⎪ p ⎩ 0, a = 0, for a ∈ GF(p). By quadratic reciprocity, if p > 2, we have ( p2 ) = (−1) p 2 −1 8 .

24 1 Background on Information Theory and Coding Theory Ideals which are of the form I = g(x)Rn are called principal ideals and g(x) is called a generator of the ideal I . Proof Suppose not. Let s(x) be a nonzero element in I of smallest degree. Pick an arbitrary nonzero element f (x) in I . By the division algorithm, we can write f (x) = q(x)s(x) + r(x), where q and r are polynomials, and the degree of r(x) is strictly less than the degree of s(x). Notice that r(x) = f (x) − q(x)s(x) belongs to I by definition.

### A survey of the Hodge conjecture by James D. Lewis

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