Computing with Riemann Surfaces and Abelian Functions

41 Pages·2014·0.49 MB·English

Computing with Riemann Surfaces and Abelian
Functions
General Examination
Chris Swierczewski
University of Washington
Department of Applied Mathematics
March 14, 2014
Abstract
The goal of my research is to use the theory of Riemann surfaces and
Abelianfunctionstoaddressdifferentapplicationproblemsandtodevelop
the software tools necessary for computing with these objects.
Abelianfunctionsareperiodicfunctionsofncomplexvariableshaving
2n independent periods. Although Abelian functions first arose in the
studyofAbelianintegrals, theyfindapplicationinmanyfieldsofmathe-
maticssuchassolvingnon-linearintegrablepartialdifferentialequations,
complex algebraic geometry, optimization, and more. Algebraic curves
andRiemannsurfacesformanaturalenvironmentforstudyingthesefunc-
tions.
In my general examination, I will present the basic theory and algo-
rithms involved in computing with these objects, demonstrate the imple-
mentation of these algorithms in the Python software library “abelfunc-
tions” I developed, and present my objectives for this research.
1 Introduction
TheKadomtsev-Petviashvili(KP)equationisapartialdifferentialequationused
todescribethesurfaceheightofatwo-dimensionalperiodicshallowwaterwave.
Depending on certain physical considerations, which we will ignore, one can
derive either of the following two equations
(−4u +6uu +u ) +3σ2u =0, σ2 =−1, (1.1)
t x xxx x yy
(−4u +6uu +u ) +3σ2u =0, σ2 =+1, (1.2)
t x xxx x yy
where u(x,y,t) is the surface height as a function of position (x,y) and time t.
In the sequel we do not rely on this distinction and we simply refer to the KP
equation.
1
TheKPequationadmitsalargefamilyofquasiperiodicsolutionsoftheform
u(x,y,t)=2∂2logθ(Ux+Vy+Wt+z ,Ω)+c, (1.3)
x 0
where θ is the Riemann theta function.
Definition 1.1. The Riemann theta function θ :Cg×h →C is defined in
g
terms of its Fourier series:
(cid:88) 2πi(cid:16)1n·Ωn+n·z(cid:17)
θ(z,Ω)= e 2 . (1.4)
n∈Zg
This function converges absolutely and uniformly on compact sets in Cg×h
g
where h is the space of all “Riemann matrices” — complex symmetric matrices
g
with positive definite imaginary part.
From the definition, we see that the Riemann theta function is periodic in
z with integer periods and quasi-periodic in z in the columns of Ω. In other
words, if m,n∈Zg then
(cid:16)1 (cid:17)
−2πi n·Ωn+n·z
θ(z+m+Ωn,Ω)=e 2 θ(z,Ω). (1.5)
AgeneralizationoftheRiemannthetafunction,involvinganon-integershift
in some of its arguments, is referred to as the Riemann theta function with
characteristics.
Definition 1.2. Let α,β ∈ [0,1)g. The Riemann theta function with
characteristic [α] is defined as
β
(cid:20)α(cid:21) (cid:88) 2πi(cid:16)1(n+α)·Ω(n+α)+(n+α)·(z+β)(cid:17)
θ (z,Ω)= e 2
β
n∈Zg
(cid:16)1 (cid:17)
2πi α·Ωα+α·(z+β)
=e 2 θ(z+Ωα+β,Ω).
Note that θ[0](z,Ω) = θ(z,Ω). See [4, 16, 17] for further definitions and
0
properties of the Riemann theta function.
Thesesolutions(1.3)aretheso-calledfinite genus solutionstotheKPequa-
tion and families of such solutions exist for every g >0. In fact, the totality of
solutions of this form are dense the space of all periodic solutions to KP [13].
The constants c ∈ C, U,V,W,z ∈ Cg and Ω ∈ h , as well as the genus g, are
0 g
determined from a Riemann surface. Any such Riemann surface can produce a
family of solutions to KP [7]. We postpone the definition of these constants in
terms of known quantities until more machinery is developed in the following
sections.
In general, periodic solutions to integrable partial differential equations are
Abelian functions.
2
Definition 1.3. An Abelian function f : Cg → C of genus g ≥ 1 is a
meromorphicfunctionsuchthatthereexists2g vectorsw ,...,w ∈Cg linearly
1 2g
independent over the real numbers where
f(z+w)=f(z),
for all z ∈Cg. When g =1 these are the elliptic functions.
Abelian functions first arose from the study of Abelian integrals
(cid:90) z1 P(z,w)
dz,
Q(z,w)
z0
whereP,Q∈C[z,w]andz andwarerelatedbyanalgebraicequationf(z,w)=
0 with f ∈C[z,w].
RiemannthetafunctionsplayacentralroleinthetheoryofAbelianfunctions
inthatallAbelianfunctionscanbewrittenasarationalfunctionoftheRiemann
theta function and its derivatives (such as in the KP solution above). The
primaryfocusofmystudyistheconstructionandnumericalevaluationofthese
Abelian functions, particularly those arising in applications.
Abelian functions are applicable in fields other than nonlinear water waves.
Forexample,theymakeexplicitmanycomputationssuchasthoseinthestudyof
solitarywaves,blackholespace-times,andalgebraiccurves. Onebasicexample
is the calculation of bitangent lines of plane algebraic curves; these are useful
for computations in optimization-related fields such as algebraic geometry and
convex optimization. Bitangents can used to represent smooth complex plane
quartic curves as either a symmetric determinant of a linear form or as a sum
of three squares [22]. In convex optimization, bitangents are used to construct
a visibility complex which, in turn, is used to solve the shortest path problem in
Euclidean space [24].
Definition 1.4. A bitangent to a plane algebraic curve C : f(x,y) = 0,f ∈
C[x,y] is a line L⊂C that lies tangent to C at at least two distinct points.
By Bezout’s Theorem, if a curve has a bitangent it necessarily must be
of degree at least four [3]. A result of Plu¨cker determines that a degree four
complexcurveadmitsexactly28complexbitangents[23]. Inparticular,Plu¨cker
showed that the number of real bitangents of any real quartic must be 28, 16,
or fewer than 9. The connection between Riemann theta functions and the
bitangent lines of smooth quartics was known to Riemann [2, 26] and, in fact,
can be computed using the tools developed in this research. See Figure 1.1 for
an example.
Finally, Riemann theta functions and algebraic curves can be used to com-
putelinearmatrixrepresentationsofalgebraiccurves. Atheoremfromclassical
algebraicgeometrystatesthateveryhomogenouspolynomialf ∈P2C[x ,x ,x ]
0 1 2
can be written in the form
f(x ,x ,x )=det(Ax +Bx +Cx ),
0 1 2 0 1 2
3
Figure 1.1: The real graph of the Edge Quartic C :f(x,y)=25(x4+y4+1)−
34(x2y2+x2+y2)=0 (in blue) and its 28 real bitangents (in grey). Note that
four of them lie tangent to C at infinity. These lines were computed using the
Riemann theta function.
where A,B,C are symmetric complex matrices which can be efficiently com-
puted using Riemann theta functions. Furthermore, when the polynomial has
real coefficients then A,B,C are symmetric real matrices and such represen-
tations are important in the study of spectrahedra — the solution spaces of
semidefinite programs [21].
Thepurposeofmyresearchistodevelopefficientandperformantalgorithms
forcomputingwithAbelianfunctionsonRiemannsurfaces. Thecomputational
tools developed in this research program have far-reaching and varied applica-
tions.
2 Complex Algebraic Geometry
This section serves as a brief introduction to the theory of complex algebraic
curves. Primary references are [10, 29].
2.1 The Projective Line
Theprimarymotivationbehindcomplexprojectivegeometryistomakeconcrete
the way in which we analyze the behavior of functions, such as polynomials, at
infinitywithouthavingtoresorttotechniquesseparatefromthoseusedatfinite
points. Forexample,inapplicationswemayneedtointegrateadifferentialalong
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a path on an algebraic curve going to infinity. Knowing the geometry of the
curve at infinity makes such an operation computationally feasible.
In fact, anyone with an elementary complex analysis background has seen
an example of projective geometry. The Riemann sphere is the complex plane
C with a “point at infinity” added. Let z denote the coordinate in C (i.e., the
point z =0 represents the origin of the complex plane). In order to discuss the
point at infinity we introduce the coordinate w = 1/z. The analysis of some
function at ∞ is equivalent to rewriting the problem in terms of the coordinate
w and examining its behavior in a neighborhood of w = 0. This explains why,
for example, the exponential function
∞
(cid:88)
ez = zn/n!,
n=0
thoughentireinthecomplexplane,hasanessentialsingularityontheRiemann
sphere since the exponential function in the coordinate w centered at w = 0 is
expressed by the series
(cid:88)∞ w−n
.
n!
n=0
This point at infinity is not rigorously defined because it does not make
sense to equate z = ∞. The definition of the Riemann sphere is made explicit
by the following construction: consider the set U = C2−{(0,0)}. Define the
equivalence relation
(a ,a )∼(λa ,λa ), ∀λ∈C−{0}.
0 1 0 1
Thus two points (a ,a ) and (b ,b ) in U are considered the same if the ratios
0 1 0 1
a : a and b : b are equal. The set of all points (b ,b ) equal to (a ,a ) is
0 1 0 1 0 1 0 1
calledtheequivalence classof(a ,a )andthecomplex projective lineP1Cisthe
0 1
set of all such equivalence classes. That is,
P1C:=C2/∼.
The equivalence class of (a ,a ), called a “point” in P1C, is written (a :a )∈
0 1 0 1
P1C. P1CispreciselytheRiemannsphere. Toseethis,considerthetwosubsets
U ={(a :a )∈P1C |a (cid:54)=0},
0 0 1 0
U ={(a :a )∈P1C |a (cid:54)=0}.
1 0 1 1
For any (a :a )∈U we have, by the equivalence property,
0 1 0
(a :a )=(1:a /a )=(1:a).
0 1 1 0
Similarly,(b :b )=(b:1)foreverypointinU . Everypointintheintersection
0 1 1
U ∩U can be written in either of these two forms. Each of these subspaces
0 1
are isomorphic to C since the maps
φ :U →C, φ ((a :a ))=a /a , and
0 0 0 0 1 1 0
φ :U →C, φ ((a :a ))=a /a ,
1 1 1 0 1 0 1
5
are continuous bijections with inverses
φ−1(a)=(1:a), (2.1)
0
φ−1(b)=(b:1). (2.2)
1
Finally, note that (0 : 1) is the only projective point in U which is not in U .
1 0
Therefore, we identify U with the complex plane (in the coordinate z) and the
0
point P =(0:1) with the point at infinity and set
∞
P1C=U ∪{(0:1)}∼=C∪P . (2.3)
0 ∞
Indeed P is considered the point at infinity on the Riemann sphere for if
∞
oneconsiderstheimageof(0:1)underφ ,thoughundefinedsince(0:1)(cid:54)∈U ,
0 0
itmapstoz =1/0“=”∞. Again,thisdoesnotmakesensewithoutthecomplex
projective space construction above but is merely used to illustrate the point.
The coordinate transformation from z to w at the beginning of this section is
equivalent to identifying U with the complex plane C and {(1 : 0)} with the
1
point at infinity, instead.
2.2 The Projective Plane
Thenaturalenvironmentweuseinthesequelisnotthecomplexprojectiveline
but the complex projective plane. In this section we construct the projective
plane and examine its geometric properties. The construction is similar to that
of the projective line.
Let U = C3−{(0,0,0)}. Following the strategy of the previous section,
considerthesetofallratios(a :a :a ),thatis,thecollectionofallequivalence
0 1 2
classes under the equivalence relation (a : a : a ) ∼ (λa : λa : λa ),∀λ ∈
0 1 2 0 1 2
C−{0}. The space of all such equivalence classes is called the two-dimensional
complex projective space or the projective plane and is denoted P2C.
Define the subsets U ,U ,U by
0 1 2
U ={(a :a :a )∈P2C |a (cid:54)=0},
j 0 1 2 j
and note that all (a : a : a ) ∈ U satisfy (a : a : a ) = (1 : a /a : a /a ).
0 1 2 0 0 1 2 1 0 2 0
We define the bijective mapping
φ :U →C2,
0 0
(cid:18) (cid:19)
a a
φ ((a :a :a ))= 1, 2 ,
0 0 1 2 a a
0 0
φ−1((x,y))=(1:x:y).
0
Themappingsφ andφ aresimilarlydefinedonU andU ,respectively. There-
1 2 1 2
fore, we can identify U with the complex plane C2.
0
Consider the space Uc = P2C−U . By definition, every point in Uc is of
0 0 0
the form (0 : a : a ). By definition, every point in Uc determines a point on
1 2 0
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the complex projective line P1C. The converse is true as well, resulting in the
bijection
(0:a :a )∈P2C ↔ (a :a )∈P1C.
1 2 1 2
By identifying Uc with P1C we may write
0
P2C=U ∪Uc ∼=C2∪P1C (2.4)
0 0
whereUc ∼=P1Ciscalledtheline at infinity,denotedl ,andU ∼=C2 iscalled
0 ∞ 0
the complex affine plane. We may also identify the complex affine plane with
the sets U or U and the line at infinity with their complements.
1 2
We saw a natural geometric interpretation of P1C in the previous section.
DoessuchaninterpretationexistforP2C? Consideralineinthecomplexaffine
plane C2 which can be written in the form
α+βx+γy =0, where (β,γ)(cid:54)=0,α,β,γ ∈C.
Using the inverse mapping φ−1 on C2 we have
0
x x
x= 1 and y = 2,
x x
0 0
where (x :x :x ) are the coordinates of P2C, and we get the line
0 1 2
αx +βx +γx =0.
0 1 2
This equation, called the homogenization of the affine curve, makes sense in all
of P2C. Setting x =1 gives the original affine line. On the other hand, setting
0
x =0 gives the equation
0
βx +γx =0,
1 2
which is the equation of the line in l . However, this implies x /x = −γ/β.
∞ 1 2
Hence the projective point (0:−γ :β) satisfies the equation
αx +βx +γx =0
0 1 2
and is, in fact, the only projective point in l on the line.
∞
This means that the line “intersects” l at the point (0 : −γ : β) and that
∞
thisintersectionpointdependsonlyontheslopeoftheaffineportionoftheline.
Hence, the line at infinity has the geometric meaning that each point on it is
the intersection point of an entire family of parallel lines in C2. This leads to
a generalization of a theorem from classical planar geometry: any two, distinct
lines in P2C intersect at exactly one point.
2.3 Projective Plane Curves
The set of all points (x ,x ,x ) satisfying
0 1 2
αx +βx +γx =0
0 1 2
7
is called a projective line and is a simple example of a projective algebraic
curve (of degree one). In this section we introduce various properties of general
projective curves.
An complex plane algebraic curve is the zero locus of the homogenization
of a polynomial f ∈ C[x,y]. That is, given a polynomial f(x,y) = α (x)yn+
n
α (x)yn−1+···+α (x)itshomogenizationisthepolynomialF ∈P2C[x ,x ,x ]
n−1 0 0 1 2
where
F(x ,x ,x )=xdf(x /x ,x /x ).
0 1 2 0 1 0 2 0
where d is the degree of F. The homogeneity of F means that we can write
(cid:88)
F(x ,x ,x )= α xixjxk.
0 1 2 ijk 0 1 2
i+j+k=d
In terms of the projective polynomial F, its affine part can be written
f(x,y) = F(1,x,y). As in the case of a projective line, f can be thought of as
a projection of the polynomial F onto C2 and there is always a one-to-one cor-
respondence between an affine polynomial and its homogenization. Therefore,
a complex plane algebraic curve is the set
C =(cid:8)(x :x :x )∈P2C:F(x ,x ,x )=0(cid:9).
0 1 2 0 1 2
Important to the study of projective curves, and specifically in the compu-
tational work described here, are singular points.
Definition 2.1. A point a = (a : a : a ) ∈ C is a singular point of C, or
0 1 2
a multiple point of C, if
(cid:18) (cid:19)
∂F ∂F ∂F
, , (a)=(0,0,0).
∂x ∂x ∂x
0 1 2
Consider the case when a = (1 : 0 : 0) (corresponding to the point (0,0) in
the affine plane C2) is a singular point of F. The affine poirtion of the curve is
d
(cid:88)
f(x,y)= c xiyj.
ij
i+j≥2
Note that the constant term is zero since (0,0) is a point on the affine curve
and the linear term vanishes since (0,0) is a singular point. We write
f(x,y)=f (x,y)+f (x,y)+···+f (x,y), m≥2,
m m+1 d
where each f is the sum of all terms of f of degree n; that is, terms of the
n
form c xiyj such that i+j = n. The smallest such m with non-zero term f
ij m
appearing in f is called the multiplicity of the singular point (1 : 0 : 0). Sin-
gularities with multiplicity two are called double points, those with multiplicity
three are called triple points, and so on.
8
The homogeneous term f can be factored into linear factors
m
m
(cid:89)
f (x,y)= (α x−β y).
m j j
j=1
We call the space f (x,y) = 0 the tangent cone of the plane curve C at a =
m
(1:0:0) consistsing of a finite number of intersecting lines L :α x−β y.
j j j
When a generic affine point a = (1 : c : d) is a singular point we write the
affine curve in the form
d
(cid:88)
f(x,y)= c˜ (x−c)i(y−d)j
ij
i+j≥2
whichwecanwriteasasumofpolynomialsg (x−c,y−d)homogenousinx−c
n
and y−d.
In the case when the singular point a=(0:1:b)∈l we repeat the above
∞
process with the affine curve
1 x x
g(u,v)= F(x ,x ,x )=F(u,1,v), u= 0,v = 2,
xd 0 1 2 x x
1 1 1
which is a projection of F onto U ∼= C instead of U . We write g as a sum of
1 0
terms of the form g ui(v−b)j. Finally, in the case a=(0:0:1)∈l we use
ij ∞
the affine curve
1 x x
h(w,z)= F(x ,x ,x )=F(w,z,1), w = 0,z = 1,
xd 0 1 2 x x
2 2 2
and write h as a sum of terms of the form h wizj.
ij
Example 2.2. Consider the cubic curve
C :F(x ,x ,x )=x4x3+2x3x3x −x7
0 1 2 0 2 0 1 2 1
In complex affine space x =1 this curve is
0
f(x,y)=F(1,x,y)=y3+2x3y−x7.
A plot of f for x,y real is shown in Figure 2.1. For a=(a :a :a ) we have
0 1 2
∂F
(a)=4a3a3+6a2a3a ,
∂x 0 2 0 1 2
0
∂F
(a)=6a3a2a −7a6,
∂x 0 1 2 1
1
∂F
(a)=3a4a2+2a3a3. (2.5)
∂x 0 2 0 1
2
9
f(x,y)=0
1
y
0
−1
−1 0 1
x
Figure 2.1: A real plot of the curve C : f(x,y) = y3 +2x3y −x7. The plot
suggests that (x : x : x ) = (1 : 0 : 0), corresponding to (x,y) = (0,0), is a
0 1 2
singular point of C.
First, we find the finite singular points of C. Setting a = 1 and solving
0
the above equations for a and a we see that p = (1 : 0 : 0) is the only finite
1 2
singular point of F. Note that
f(x,y)=f (x,y)+f (x,y)+f (x,y),
3 4 7
f (x,y)=y3, f (x,y)=2x3y, f (x,y)=−x7,
3 4 7
and that f , f , and f are homogeneous of degrees 3, 4, and 7, respectively.
3 4 7
Therefore, p is a singular point of multiplicity 3 with
f (x,y)=y3 =0,
3
as the equation for the tangent cone at p. These properties are suggested by
Figure 2.1 where, near the point p, the real curve looks like the intersection of
three curves well approximated by the line y =0 near the point x=0.
Setting a = 0, the only expression in Equation (2.5) that does not reduce
0
to zero is
∂F
((0,a ,a ))=−7a6 =0,
∂x 1 2 1
1
implying that the point a=(0:0:1) is the only singular point at infinity. The
curve at infinity centered at (0:0:1) is
h(w,z)=F(w,z,1)=w4+2w3z3−z7.
10