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Results tagged with quadratic-forms
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user 6214
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
15
votes
Accepted
Why are there usually an even number of representations as a sum of 11 squares
Throughout $N>0,$ and $N \equiv 3 \pmod 8.$ Let $I$ be the number of ordered triples $(a,d,e) \;\mbox{with} \; a,d,e \geq 0,$ such that
$$a^2+2 d^2+8 e^2=N.$$ I'll use a result of Gauss on sums of 3 …
- 5,422
5
votes
Accepted
Primes and $x^2+2y^2+4z^2$
Here's a simpler argument. We may assume p is 7 mod 8. Let N be the number of triples of squares (r,s,u) with r+2s+4u=p. We will show that N is odd if p is 7 mod 16 and even if p is 15 mod 16. Let M b …
- 5,422
6
votes
Accepted
Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4?
This is nothing like a complete answer (EDIT-now it seems to be--see below), but it may suggest a fruitful attack, based on the comment (see below for a version incorporated into this answer) that I m …
- 5,422
1
vote
Accepted
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb...
(Part 1)--My argument uses the following curious fact about ideals in $Z[i]$ and $Z[\sqrt{-2}].$
Suppose $n=8m+1$. Let $I=I(n)$ and $J=J(n)$ be the number of ideals of norm $n$ in $Z[i]$ and $Z[\sqrt{ …
- 5,422
1
vote
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb...
Part 2--the curious fact
The theory of quadratic fields tells us that I is the sum of the Jacobi symbols (-1/d) and
J is the sum of the (-2/d) where d divides n. Write n as a product of powers of dis …
- 5,422
2
votes
More questions involving characteristic 2 theta series identities
I can now, with less computer calculation than I'd feared, answer Question 1. (I'll say more about Question 2 later).
Lemma:__ Let V be the vector space over Z/2 spanned by the C(r1,r2,r3) and the C( …
- 5,422