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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
n&= #13;
n&= #13;
nReal dimension ≤ complex dimension
y
x
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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
y
x
C1
y
x
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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;
n
------=_NextPart_01C6DF37.43FB3610 Content-Location: file:///C:/8D8AE513/Wampler_RealCurves_files/slide0163.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Finding All Real Points of a Complex Curve
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
------=_NextPart_01C6DF37.43FB3610 Content-Location: file:///C:/8D8AE513/Wampler_RealCurves_files/slide0157.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Finding All Real Points of a Complex Curve
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
n
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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
n
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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
------=_NextPart_01C6DF37.43FB3610 Content-Location: file:///C:/8D8AE513/Wampler_RealCurves_files/slide0167.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Finding All Real Points of a Complex Curve
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
3D"Line
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
3D"Rounded
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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R0lGODlhagA8AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABn ADoAgAAAAAAAAALHhI8Yy+0Po5w0povTy7z7D3obyITmiaZXmS3qC8edq8n2bQcKzveoDgD6hsRW sIhMHpVMorAJzUWnsif1arJitzOuV/QNY7TiL7nMPaOx6jW17Y7C4805XWm/I/N6Z3/L99cTKIhD WCiFKKe4yFjn+AiJJzlJuWd5iemnOXTI+eH52SXKE0o6ZnqqkKq6xEr69MqpRdNKwlFru5LbwtLa oFIhPFxR5SB6nMKrOkIiW5g886wYvTvN2HywrIsLbLDNLV0dfuJSAAA7 ------=_NextPart_01C6DF37.43FB3610 Content-Location: file:///C:/8D8AE513/Wampler_RealCurves_files/slide0172_image083.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbQBFAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABr AEQAgAAAAAAAAAL/hI+py+1/gpxtSoiz3vomHwUfyJXmY5GRIwLqd8ay8VI09C7tzGM5ItqxOr1i JSMcEo2mFCmpRHGgTFbOQ2VkedsqENf1HsVjH3ljO9PCip/663rH5We2zE6/k/F5vZffFwNoFvg2 iFMod1iWWGe02Ij2GKkGqUPpWGSJKdXjxul3FwRatXmzQqo5Y5qqNdfUKtayyBrbhjplyyREq5u5 5CubGywsSfw3fLxbUquMC+zMBRvdBthLfUk4jQ2DKNjcGGYalEYtHsrdnS2Yvq7O3n77fgLOmVRb 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Foldable Platform: Step 1
nNumerical Irreducible Decomposition gives
n1 double quadric surface
nNonphysical, throw away
n12 nonphysical lines (throw these away)
n3 double lines (deflate these)
n3 quadric curves
n4 quartic curves
nRemark: 2x3+3x2+4x4=3D28
Find real points
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Foldable platform (cont)
nAll 10 components are self-conjugate
nEach of the 3 lines are completely determined by their witness point and the tangent at it. =
n= Do nothing more to these
nRun the algorithm on the other 7 pieces
n= 3 real points show up in the B* set for several components
nEach quadric passes through 2 of these =
n= Three quartics have:
nno real B*
nno real points on a random real slice <= /span>
nhence no real points.
n= The other quartic has a double point in its B* set=
n
------=_NextPart_01C6DF37.43FB3610 Content-Location: file:///C:/8D8AE513/Wampler_RealCurves_files/slide0175.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Finding All Real Points of a Complex Curve
The special B* points
nAn example of how a numerical result can lead one = to see an exact result
n= (The numerics were done to 12 digits. Only 4 are s= hown.)
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1EUrCQNfVEbR6Gu4cEZvVJFiFEd3tG4mQgN+VOt4VEjPR0eH1EiPFEnpqUiTlEmb1Em7aUmfVEqn lEqNKUqrFEuzVEtX6Uq31Eu/FExzVETDdEgjAAA7 ------=_NextPart_01C6DF37.43FB3610 Content-Location: file:///C:/8D8AE513/Wampler_RealCurves_files/slide0176.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Finding All Real Points of a Complex Curve
A projection of the real curves
Curves live in P7.
nProject onto a C7 patch,
nthen project down to C2.
Key:
  =3D B*
  =3D B\B* = ;
  =3D edge point
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