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4 votes
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216 views

Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism

Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
wlad's user avatar
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2 votes
0 answers
114 views

Existence theorem for symmetric nondegenerate forms over a ring

There exists a rich theory for inner product spaces (i.e. vector spaces with a symmetric nondegenerate bilinear form) over fields, and it can be discussed in the context of local rings and free ...
Plank's user avatar
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2 votes
0 answers
98 views

Classification of quadratic submodules in $\mathbb C[[t]]$

Let $\mathbb C[[t]]$ be the ring of formal series with complex coefficients. Let $M$ be a finite rank free module over this ring. Let $Q$ be a regular quadratic form on $M$. (E.g., the standard ...
Roman Fedorov's user avatar
1 vote
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28 views

Modified straightline complexity of almost square of sums

Assume every linear operation (such as inner product with constant vector) can be performed in one step and multiplication by variables (quadratic operation) can be performed in one step. We know the ...
VS.'s user avatar
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1 vote
0 answers
295 views

Proper ideals are invertible

I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma: Lemma 7.5: Let $...
user50139's user avatar
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0 votes
0 answers
111 views

Totally isotropic space for bilinear pairing over ring

A duplicate of this: Consider the following well-known inequality: Let $b$ be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$ a totally ...
JBuck's user avatar
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0 votes
0 answers
94 views

Tensor product of preordered rings

All rings in this post are commutative, unital, and contain $\frac{1}{2}$. To study "real" properties of a ring $R$, one is often interested in the orderings which exist on fraction fields of ...
Bib-lost's user avatar
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0 votes
1 answer
213 views

number of representations by sums of three squares (with coefficients)

There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for $$ \#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\} ?$...
Dr. Pi's user avatar
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