binary quadratic forms (bqf)tghyde/hyde -- exploring the topograph.pdf · binary quadratic forms...
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Binary Quadratic Forms (BQF)
I Binary quadratic form:
Degree 2 homogeneous polynomial in 2 variables.
f(x, y) = ax2 + bxy + cy2
I Ex. f(x, y) = 3x2 − 2xy + y2.
I Seems easy?
![Page 3: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/3.jpg)
Binary Quadratic Forms (BQF)
I Popular with old-timey Europeans like Fermat, Lagrange,Legendre, Euler, Gauss.
I Classic questions:I Which primes are the sum of two squares?
p = x2 + y2?
I When is a BQF invariant under change of coordinates?
f(x, y) = x2 − xy + y2 = f(x− y, x).
I When are two BQF the same after change of coordinates?
![Page 4: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/4.jpg)
What’s the Big Deal?!
Legendre writing aboutBQF in 1798.
David A. Cox writing aboutBQF in 1989.
![Page 5: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/5.jpg)
What’s the Big Deal?!Conway writing about BQF in 1997!
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Introducing: the Topograph
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Introducing: the Topograph
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Introducing: the Topograph
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Introducing: the Topograph
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Topographical FactsI No vector ever repeats.I Vectors
[ab
]and
[cd
]are adjacent if and only if
ad− bc = det
[a cb d
]= ±1.
![Page 11: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/11.jpg)
Topographical Facts
Say[ab
]is primitive if gcd(a, b) = 1.
I Every primitive vector shows up in the topograph.
I Define the Top and Bottom operations by
T[ab
]=
[a + bb
]B[ab
]=
[a
a + b
].
I Every primitive vector can be made from[10
]or
[01
]using a
sequence of top and bottom operations.
![Page 12: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/12.jpg)
Topographical Facts
Idea: work backwards![311
]= B3
[32
]= B3T1
[12
]= B3T1B2
[10
].
Can you see why this works? Try more examples!
![Page 13: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/13.jpg)
Topographical Facts
Idea: work backwards![311
]= B3
[32
]= B3T1
[12
]= B3T1B2
[10
].
Can you see why this works? Try more examples!
![Page 14: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/14.jpg)
BQF on the Topograph
f(x, y) = ax2 + bxy + cy2
I f(kx, ky) = a(kx)2 + b(kx)(ky) + c(ky)2 = k2f(x, y)
I f(−x,−y) = f(x, y)
I Enough to know the values of f(x, y) on primitive vectors.
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BQF on the Topograph
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BQF on the Topograph
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The Arithmetic Progression Law
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From (Almost) Nothing to Everything!
I We don’t even need to know the vectors!I Coordinate free way to study BQF.
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From (Almost) Nothing to Everything!
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From (Almost) Nothing to Everything!
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From (Almost) Nothing to Everything!
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From (Almost) Nothing to Everything!
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Formula Found
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The Climbing Lemma
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The Climbing Lemma
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Race to the Bottom
I We can always head downhill!
I What happens at the bottom?
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Stuck in the Well
I Climbing Lemma: At most one well.I If f > 0, then it has a well and f is called positive definite.
I To tell if two positive definite forms are equivalent, go meetat the well.
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Down by the River
I Climbing Lemma: At most one river!I If f 6= 0 and has a river, then f is called indefinite.
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Down by the River
I Climbing Lemma: At most one river!I If f 6= 0 and has a river, then f is called indefinite.
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DiscriminationIf a and c are adjacent values of f with common difference hbetween them, then the discriminant of f is
∆ = h2 − 4ac.
I Same value measured anywhere on the topograph!I Ex. ∆ = −8.
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River or Well?
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The Repeating River
Suppose f has a river.
I Finitely many options for a, c, h.
I Therefore the river must always repeat!
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The Repeating River
f(x, y) = x2 + 3xy − 2y2
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Lakes
I If f has a lake, then ∆ = h2 is a square.I One lake =⇒ f(x, y) = ax2.
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Lakes
If both ± appear, there are two lakes connected by a river.
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Lakes
Sometimes the river floods and the lakes merge.
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Symmetries
I A symmetry of a BQF f is a linear change of coordinatesthat leaves f invariant,
f(Ax + By,Cx + Dy) = f(x, y).
I Alternatively, it is a transformation of the topograph thatpreserves the f labelling.
![Page 38: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/38.jpg)
SymmetriesI A symmetry of a BQF f is a linear change of coordinates
that leaves f invariant,f(Ax + By,Cx + Dy) = f(x, y).
I Alternatively, it is a transformation of the topograph thatpreserves the f labelling.
![Page 39: Binary Quadratic Forms (BQF)tghyde/Hyde -- Exploring the topograph.pdf · Binary Quadratic Forms (BQF) I Popular with old-timey Europeans like Fermat, Lagrange, Legendre, Euler, Gauss](https://reader035.vdocuments.site/reader035/viewer/2022063007/5fb9a7a2771ce93aa6454b42/html5/thumbnails/39.jpg)
SymmetriesSymmetries preserve all special features!
I Wells, rivers, and lakes, are preserved.
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Another Perspective
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Another Perspective
The mediant of reduced fractions ab and c
d is a + cb + d .
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Another Perspective
The mediant of reduced fractions ab and c
d is a + cb + d .
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Equivalence
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Old Facts, New Light
Two rational numbers ab and c
d are adjacent if and only if theirdifference is an Egyptian fraction 1
m .
ab −
cd =
ad− bcbd = ± 1
bd .
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Old Facts, New Light
Every reduced fraction eventually appears.I Define Top and Bottom operations on fractions by
T(ab
)=
a + bb B
(ab
)=
aa + b .
I Notice: T is “add one” and B is “flip, add one, flip.”
T(ab
)= 1 +
ab B
(ab
)=
1a+ba
=1
1 + ba
=1
1 + 1ab
.
I Applying T and B repeatedly is easy.
Tm(ab
)= m +
ab Bm
(ab
)=
1m + 1
ab
.
I (flip, add one, flip)(flip, add one, flip) = (flip, add two, flip)
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Continued Fractions
I Recall that [311
]= B3T1B2
[10
].
I Check it out:
B3T1B2( 10
)=
1
3 +1
1 +1
2 +1∞
=1
3 +1
1 +12
=1
3 +23
=311
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Continued Fractions
Continued fraction expansion of ab is a path from∞ to a
b !
311 =
1
3 +1
1 +1
2 +1∞
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But wait, there’s more!
New model shows there are hidden cells in our topograph livingoff near the horizon.
√2 = 1+
1
2 +1
2 +1
2 +1. . .
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Back to the River
f(x, y) = x2 − 2y2 =⇒ fy2 =
(xy
)2− 2
Lagrange: If d ≥ 0 and a, b, d are integers, then the continuedfraction expansion of a + b
√d is eventually periodic.
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Topographical Trinitarianism
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The End
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Want More?
The Sensual (Quadratic) Formby John H. Conway.
Topology of Numbersby Allen Hatcher.