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Results tagged with quadratic-forms
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user 3324
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.
5
votes
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
Friday, June 28. I found a nice exposition by David Savitt
https://pi.math.cornell.edu/~web401/steve.gauss17gon.pdf
from which this is page 32
David A. Cox, in Galois Theory, gives an account of Gau …
- 25.7k
1
vote
Automorphism groups in class sets of ternary lattices
I put lots of references at http://zakuski.math.utsa.edu/~kap/
I've got an early version working. At first I thought it would be just class number one or two.
The six coefficients $a,b,c,d,e,f$ ref …
- 25.7k
3
votes
A cubic equation, and integers of the form $a^2+32b^2$
details, details. From $x$ odd and
$$
x^4-32x-16 = (x^2 + 4)^2 - 2(2x+4)^2.
$$
we see that $x^4-32x-16$ is not divisible by any prime $q \equiv 3,5 \pmod 8.$ That is, $x^2 + 4$ is also odd. Nex …
- 25.7k
2
votes
Integers $8k+3>0$ not represented by $2x^2+4y^2+4yz+9z^2$ over the integers
First things first, all the relevant numbers really are represented by $x^2 + 2 y^2 + 32 z^2.$ See http://zakuski.math.utsa.edu/~kap/Kap_Jagy_Schiemann_1997.pdf
for regular ternaries.
From my giant …
- 25.7k
3
votes
Integers representable as binary quadratic forms
you don't seem to be mentioning Gauss composition. You have a genus of forms, equivalent to $\langle 1,8,27 \rangle,$ then $\langle 3,8,9 \rangle,$ then $\langle 9,8,3 \rangle.$ These are convenient …
- 25.7k
0
votes
Binary quadratic forms order four in the form class group not having desired coefficients
Finally made an exhaustive program, finds stubborn order four forms; ran it up to absolute value of discriminant 2500.
144: < 5, 4, 8> STUBBORN 144: < 4, 0, 9> 144: < 1, 0, 36> 144 …
- 25.7k
4
votes
Modular forms and number of representations by binary quadratic forms
The coefficient $w$ in theorem 64 is usually $2,$ but is $4$ for discriminant $-4,$ then $6$ for discriminant $-3$
- 25.7k
2
votes
Proof that $x^2 + y^2 - z^2$ is universal
As the others point out, there is no knowing about the earliest this was written down. For example, the notion of regularity of a ternary form is due to Dickson, but universality is an easier concept …
- 25.7k
1
vote
Solving a pair of ternary quadratic form equations
Part of this is finding the primitive integer null vectors of an indefinite ternary quadratic form $f(x,y,z).$ Mordell points out that these occur in a finite number of parametrizations. The process y …
- 25.7k
0
votes
Simple conjecture about rational orthogonal matrices and lattices
This is from papers about 1940 by Gordon Pall, one with B. W. Jones. I'm looking for statements about things being primitive, especially odd/even. Found it, also in "Rational Automorphs," in order to …
- 25.7k
4
votes
When does $axy+byz+czx$ represent all integers?
I have figured out some things; it is much quicker, as far as computing, to find a way for the Hessian matrix of the ternary quadratic form, is to have it represent the (two by two) Hessian of the for …
- 25.7k
13
votes
When does $axy+byz+czx$ represent all integers?
Just so you know, one of Dickson's students (A. Oppenheim) finished classifying (indefinite) universal ternaries; the final family is $xy - M z^2.$ Page 161 in Modern Elementary Theory of Numbers. You …
- 25.7k
7
votes
Difference of two sums of two squares
Sure. Demand $a \geq b \geq 0$ as well as $c,d \geq 0.$
Then map
$$ (a,b,c,d) \mapsto (25a+11b+24c+13d, 11a-b+11c, 24a+11b+23c+13d, 13a + 13 c+d) $$
- 25.7k
4
votes
Evaluating a binary quadratic form at convergents
The item you want is the neighbor method, a version of continued fractions. I learned this from Buell, Binary Quadratic Forms. It is also in a 1929 Introduction by Dickson, and a book by Matthews I've …
- 25.7k
1
vote
Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
well, here is the same information when the target number has a factor of 2 or 3. The ratio pair (12,4) is evidently repeated, the two cases are $1,10 \pmod {12}.$
Oh, note that multiples of $9$ are …
- 25.7k