Wampler: Finding All Real Points of a Complex Curve

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Finding All Real Points of a Complex Algebraic Curve

Charles Wampler

General Motors R&D Center

In collaboration with

Ye Lu (MIT), Daniel= Bates (IMA), &

Andrew Sommese (University of Notre Dame)

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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Finding All Real Points of a Complex Curve

2&#= 13;

Outline

nReal points in a curve

nRelati= onship between a complex curve and its real points

nMorse-like representation for real curves

nNumerical algorithm

nFind isolated real points

nFind boundary points, B, of real curve arcs

nFind arcs and how they connect to B

nFind how components meet each other

nExamples

nA Griffis-Duffy platform with a curious real motion

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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Finding All Real Points of a Complex Curve

Real Solutions & Complex Dimens= ion

nAt complex dimension 0, a point is either real or not

n= Just check the imaginary part

nA complex curve may contain

n= Real arcs

n= Isolated real points

n= Example:

n[x(x-1)(x-2)]2+[y(y-1)y-2)]2=3D0

n9 isolated points in one complex curve =

n= Example: an isolated and a 1-dim real pieces in th= e same complex curve

ny2-x2<= /span>(x-1)=3D0

nReal dimension ≤ complex dimension

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Conjugation

nLet k : Cà C be the conjugation operation

n= x is real iff k(x)=3Dx

nComplex conjugation conj(a+bi)=3Da-bi <= /span>

n= Real part of V(xy-1) is a unit hyperbola

nOther conjugation operations

n= Hermitian transpose of a square matrix, A*

nk(x,y)=3D(conj(y),conj(x))

n“real” part of V(xy-1) consists of pai= rs of complex conjugate points = on the unit circle in C1

nAlgorithms require only minor adjustments for different conjugation operators =

n= i.e., sweep with a “real” hyperplane

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Conjugate components

nLet f(x) be polynomial system with real coefficients

nIf f(z)=3D0, then f(= k(z))=3D0

nSuppose f-1(0) has several 1-dimensional irreducible components

nThe components must either:

nBe self-conjugate, k= (Z)=3DZ, or

nAppear in conjugate pairs, k(Z)=3DZR= 17;≠Z&#= 13;

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Real points of conjugate pairs

nIf k(Z)=3DZ’≠Z, the real points of Z and Z’ must be their intersection ZÇZ’

nThese points must be isolated

nWe can numerically intersect any two algebraic sets

nUse a diagonal homotopy= 3;

nThis is all we need= to handle conjugate pairs.

nSelf-conjugate components are another matter

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Local smoothness

nLet Z be

n= Quasiprojective complex algebraic set <= /span>

n= Reduced

n= Dimension 1

nLet sing(Z) be its singular points

nLet reg(Z)=3DZ\sing(Z)

nSuppose zÎreg(Z) is real,

n<= span style=3D'font-size:86%'>Then there is an open complex neighborhood of z = such that the real points of Z in the neighborhood= form a smooth connected curve. <= /div>

nUpshot:

n= real points =3D real smooth arcs + RNÇ(sing(Z))

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Sweeping out the curve

nPick a general real projection p

nV(p(x)-t) as t varies along R is a “sweep hyperplane”

nFor general t, V(p(x)-t) Ç Z consists of isolated regular <= span style=3D'font-size:75%'>points

n<= span style=3D'font-size:71%'>Witness set of size d=3Ddeg Z

n= On an open real interval of t, we can numerically = sweep out arcs

n<= span style=3D'font-size:71%'>Sweep fails where:

nTangent to curve lies inside the sweep plane

nJacobian matrix at point on curve is corank = 805; 2

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Morse-like representation

nLet B* be the real points where the sweep will fail

n= Note: sing(Z) Ì B*

nLet B =3D pZ-1(p(B*)) Ç RN

nMorse-like representation of Z consists of

n= The generic real linear projection p

n= The boundary points B=3D{B1,…,Bn},

n= The edges E=3D{e1,…,ek}, where each edge is {x,l,r}:

nx is a point on an arc of ZÇ RN<= /span>

n(l,r) are pointe= rs to B:

nBl is l= eft endpoint of the arc or l=3D-= 734;

nBr is right endpoint of the arc or r= =3D+= 734;

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Algorithm for self-conjugate curves=

= 1.Define p

2.Find B*

3.Sli= ce between B*

4.Tra= ck to find endpoints and extend B* to B

Key:

=3D B* <= /span>

=3D edge point

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Content-Type: image/gif R0lGODlhaQA6AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABn ADgAggAAAAAAAADkqDMzzP8AAAECAwECAwECAwPDOLrc/jDKSSsFOOvNu/9gKI7hQJ5oqq6Zyb5w fLpybdf0re8z7/+fHHDoExKPNiNy+VIynygndFqiWnvXLEiqzXK71i94Kh4/y+YlOn1cs4fu9y8u 59Hrujs+uQ9vAoB9RDkBBIYBgkCEhgSIiUUahYePkJGBlHaYappIepxRn4OhcKOKpXOnlal5qzue rR2vsBtCgI6zoBkBAry3uCOEvAK+v1W6wsTFQX+XyiKyxdC/0rjUs9aw2K3aqxbe3+Dh4g0JADs= ------=_NextPart_01C6DF37.43FB3610 Content-Location: file:///C:/8D8AE513/Wampler_RealCurves_files/slide0164.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Finding All Real Points of a Complex Curve

Full algorithm in outline

<= span style=3D'font-size:75%'>nFor polynomial system f(x)

<= span style=3D'font-size:71%'>1.Find the 1-dimensional components

<= span style=3D'font-size:75%'>nNumerical irreducible decomposition

<= span style=3D'font-size:75%'>nDeflate any nonreduced components

<= span style=3D'font-size:71%'>2.Test to find = each component’s conjugate

<= span style=3D'font-size:71%'>3.Intersect con= jugate pairs

<= span style=3D'font-size:75%'>nUse diagonal intersection

<= span style=3D'font-size:75%'>nResult is isol= ated real points

4.Compute the Morse-like representation for the real part of each self-conjugate component

<= span style=3D'font-size:75%'>nResult is boun= dary points B and edges E

<= span style=3D'font-size:75%'>nSome boundary = points B* may be isolated

5.(Optional) Fi= nd where components not related by conjugation meet

<= span style=3D'font-size:75%'>nLet’s lo= ok at some steps in more detail

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Determining conjugate pairs

nNumerical irreducible decomposition gives us witness sets for the components =

n= Let Z,Z’ be components

nMove the slicing plane (for all components) to the same generic real plane

n= Let WZ, WZ&= #8217; be t= he witness sets for Z,Z’

nConsidering all 1-dim components of f-1= (0), witness points must appear in conjugate pairs

nWe have k(Z)=3DZ’ iff k(WZ<= /span>)=3DWZ’

n= With probability 1

n= This identifies self-conjugacy too

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Determining B*

nLet’s consider a reduced component, Z

n= Nonreduced introduces technicalities <= /span>

nEasy and not so interesting

nLet a = be the sweep direction: p(x)=3D<= span style=3D'font-size:75%'>aTx = ;

nFind B= , a basis for the sweep tangents

n= [a B] is full rank, BT<= /span>a=3D0&= #13;

nLet J(x) be the Jacobian matrix of f(x)

nDefine h(x,y)=3DJ(x)B y

n= y parameterizes the tangent space of the sweep = ;

n= Define px and py, nat= ural projections

nB* =3D px ((Z´PN-2)ÇV(h))

n= B* are isolated

n= ((Z´P= N-= 2)ÇV(h)) can be any dimension 0,…,N-2

n= Find by a witness cascade

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Extra points in B*

= nSuppose f(x):= CN<= /span>à= ;Cm has m≥N-1 equations = ;

nFor a 1-dim’l component Z =

n= “square up” f(x) as g(x)=3DQf(x), = ;

nQ is (N-1)´m

nThis can introduce an extraneous component, Z’

n= Z,Z’ Ì V(g)&#= 13;

nWe may get extra points ZÇZ’ in B*

nNot a problem, just makes extra edges <= /span>

nDeflation of nonreduced components can similarly add extra points to B*

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Examples

nThree easy

nOne substantial

nA foldable Griffis-Duffy platform

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Example 1

nSuppose f(x)=3Dx2+y2

nV(f) is two lines

n(x,y) =3D (u,ui) and (x,y)=3D(u,-ui), uÎC <= /b>

nThese are a conjugate pair

nThe lines intersect in the real point (0,0)

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Example 2

<= span style=3D'font-size:88%'>n(0,1) is an em= bedded point in V(y-1)

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Example 3

= nA single irred= ucible self-conjugate complex curve

Double = point in B*

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Example 4: Foldable Griffis-Duffy

nGriffis-Duffy platform with

nEqual triangles

nJoints at midpoints of sides

nLeg lengths all equal to altitude of the triangles

Deg 28 irreducible motion in Study <= span style=3D'position:absolute;top:22.25%;left:45.88%;width:15.91%;height:3.25= %'>Coords

(legs not equal)

What about this one?

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