Post on 26-Feb-2022
IntroductionSquare-root Parametrization
Further Directions
Square-Root Parametrizations
Jose Manuel Garcia Vallinas / Josef Schicho
Spezialforschungsbereich F013Subproject F1303
Johann Radon Institute for Computional and Applied Mathematics (RICAM)Austrian Academy of Sciences (OAW)
Linz, Austria
Workshop on Algebraic Spline Curves and SurfacesMay 17-18, 2006, Eger, Hungary
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definitions IDefinitions IIWeierstrass Form
Definitions I
Plane Algebraic Curve
An affine irreducible plane algebraic curve over C is defined asthe set
C = {(a, b) ∈ A2(C)|f (a, b) = 0}
for a non-constant irreducible polynomial f (x , y) ∈ C[x , y ].
Singular Point
Let C be an affine plane curve over C defined by f (x , y) ∈ C[x , y ]and let P = (a, b) ∈ C . P is a singular point if and only if theorder of the first non-vanishing term in the Taylor expansion of fat P is greater than 1.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definitions IDefinitions IIWeierstrass Form
Definitions I
Plane Algebraic Curve
An affine irreducible plane algebraic curve over C is defined asthe set
C = {(a, b) ∈ A2(C)|f (a, b) = 0}
for a non-constant irreducible polynomial f (x , y) ∈ C[x , y ].
Singular Point
Let C be an affine plane curve over C defined by f (x , y) ∈ C[x , y ]and let P = (a, b) ∈ C . P is a singular point if and only if theorder of the first non-vanishing term in the Taylor expansion of fat P is greater than 1.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definitions IDefinitions IIWeierstrass Form
Definitions II
GenusLet C be an irreducible plane curve of degree n, having onlyordinary singularities of multiplicities r1, . . . , rm. The genus of C ,g(C ) , is defined as
g(C ) :=1
2[(n − 1)(n − 2)−
m∑i=1
ri (ri − 1)]
For non-ordinary singularities, the genus can be computed similarly.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definitions IDefinitions IIWeierstrass Form
Weierstrass Form
A curve C is called elliptic if and only if its genus is 1
It is known that an elliptic curve can be birationally transformed tothe form y2 = F (x), where F (x) is a square-free polynomial in xof degree 3.
A curve C is called hyper-elliptic if and only if its genus g isgreater than 1 and it can be birationally transformed to y2 = F (x),where F (x) is a polynomial in x of degree 2g + 1 or 2g + 2.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definitions IDefinitions IIWeierstrass Form
Weierstrass Form
A curve C is called elliptic if and only if its genus is 1
It is known that an elliptic curve can be birationally transformed tothe form y2 = F (x), where F (x) is a square-free polynomial in xof degree 3.
A curve C is called hyper-elliptic if and only if its genus g isgreater than 1 and it can be birationally transformed to y2 = F (x),where F (x) is a polynomial in x of degree 2g + 1 or 2g + 2.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definitions IDefinitions IIWeierstrass Form
Weierstrass Form
A curve C is called elliptic if and only if its genus is 1
It is known that an elliptic curve can be birationally transformed tothe form y2 = F (x), where F (x) is a square-free polynomial in xof degree 3.
A curve C is called hyper-elliptic if and only if its genus g isgreater than 1 and it can be birationally transformed to y2 = F (x),where F (x) is a polynomial in x of degree 2g + 1 or 2g + 2.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Square-root Parametrization Problem
Square-root Parameterisable Curve
A curve is called square-root parameterisable if it can beparameterised in terms of t and
√P(t), where P(t) is a
polynomial in t.
Example: x2 + y2 − 1: t 7→ (t,√
1− t2) or (√
1− t2, t)
ProblemWe have an algebraic (irreducible) plane curve C and we want toknow if it is square-root parameterisable and compute thisparametrization in the positive case.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Square-root Parametrization Problem
Square-root Parameterisable Curve
A curve is called square-root parameterisable if it can beparameterised in terms of t and
√P(t), where P(t) is a
polynomial in t.
Example: x2 + y2 − 1:
t 7→ (t,√
1− t2) or (√
1− t2, t)
ProblemWe have an algebraic (irreducible) plane curve C and we want toknow if it is square-root parameterisable and compute thisparametrization in the positive case.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Square-root Parametrization Problem
Square-root Parameterisable Curve
A curve is called square-root parameterisable if it can beparameterised in terms of t and
√P(t), where P(t) is a
polynomial in t.
Example: x2 + y2 − 1: t 7→ (t,√
1− t2) or (√
1− t2, t)
ProblemWe have an algebraic (irreducible) plane curve C and we want toknow if it is square-root parameterisable and compute thisparametrization in the positive case.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Square-root Parametrization Problem
Square-root Parameterisable Curve
A curve is called square-root parameterisable if it can beparameterised in terms of t and
√P(t), where P(t) is a
polynomial in t.
Example: x2 + y2 − 1: t 7→ (t,√
1− t2) or (√
1− t2, t)
ProblemWe have an algebraic (irreducible) plane curve C and we want toknow if it is square-root parameterisable and compute thisparametrization in the positive case.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Rational Curves
If we have a rational curve, we take its rational parametrization.
Example
frational := y2 − x2(x + 1);genus(frational , x , y) = 0;squareRoot(frational , x , y , t) = [−1 + t2, t(−1 + t2)]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Rational Curves
If we have a rational curve, we take its rational parametrization.
Example
frational := y2 − x2(x + 1);
genus(frational , x , y) = 0;squareRoot(frational , x , y , t) = [−1 + t2, t(−1 + t2)]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Rational Curves
If we have a rational curve, we take its rational parametrization.
Example
frational := y2 − x2(x + 1);genus(frational , x , y) = 0;
squareRoot(frational , x , y , t) = [−1 + t2, t(−1 + t2)]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Rational Curves
If we have a rational curve, we take its rational parametrization.
Example
frational := y2 − x2(x + 1);genus(frational , x , y) = 0;squareRoot(frational , x , y , t) = [−1 + t2, t(−1 + t2)]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Elliptic Curves
If we have an elliptic curve, we simply need to compute itsWeierstrass form and substitute.
Example
felliptic := x3 − y3 − x ;genus(felliptic , x , y) = 1;WeierstrassForm(felliptic) = x3 + y2 − 1
squareRoot(felliptic , x , y , t) = [√−t3+1(t+1)3+3t+3t2 ,
√−t3+1(t+2)3+3t+3t2 ]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Elliptic Curves
If we have an elliptic curve, we simply need to compute itsWeierstrass form and substitute.
Example
felliptic := x3 − y3 − x ;
genus(felliptic , x , y) = 1;WeierstrassForm(felliptic) = x3 + y2 − 1
squareRoot(felliptic , x , y , t) = [√−t3+1(t+1)3+3t+3t2 ,
√−t3+1(t+2)3+3t+3t2 ]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Elliptic Curves
If we have an elliptic curve, we simply need to compute itsWeierstrass form and substitute.
Example
felliptic := x3 − y3 − x ;genus(felliptic , x , y) = 1;
WeierstrassForm(felliptic) = x3 + y2 − 1
squareRoot(felliptic , x , y , t) = [√−t3+1(t+1)3+3t+3t2 ,
√−t3+1(t+2)3+3t+3t2 ]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Elliptic Curves
If we have an elliptic curve, we simply need to compute itsWeierstrass form and substitute.
Example
felliptic := x3 − y3 − x ;genus(felliptic , x , y) = 1;WeierstrassForm(felliptic) = x3 + y2 − 1
squareRoot(felliptic , x , y , t) = [√−t3+1(t+1)3+3t+3t2 ,
√−t3+1(t+2)3+3t+3t2 ]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Elliptic Curves
If we have an elliptic curve, we simply need to compute itsWeierstrass form and substitute.
Example
felliptic := x3 − y3 − x ;genus(felliptic , x , y) = 1;WeierstrassForm(felliptic) = x3 + y2 − 1
squareRoot(felliptic , x , y , t) = [√−t3+1(t+1)3+3t+3t2 ,
√−t3+1(t+2)3+3t+3t2 ]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have theparametrization. Simply compute the Weierstrass form of thecurve.
Example
fhyperelliptic := x4 + y4 + x2 + y2;genus(fhyperelliptic , x , y) = 2;WeierstrassForm(fhyperelliptic) = y2 + x6 + x2 + x4 + 1
squareRoot(fhyperelliptic , x , y , t) = [ t2+1√−t6−t4−t2−1
, (t2+1)t√−t6−t4−t2−1
]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have theparametrization. Simply compute the Weierstrass form of thecurve.
Example
fhyperelliptic := x4 + y4 + x2 + y2;
genus(fhyperelliptic , x , y) = 2;WeierstrassForm(fhyperelliptic) = y2 + x6 + x2 + x4 + 1
squareRoot(fhyperelliptic , x , y , t) = [ t2+1√−t6−t4−t2−1
, (t2+1)t√−t6−t4−t2−1
]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have theparametrization. Simply compute the Weierstrass form of thecurve.
Example
fhyperelliptic := x4 + y4 + x2 + y2;genus(fhyperelliptic , x , y) = 2;
WeierstrassForm(fhyperelliptic) = y2 + x6 + x2 + x4 + 1
squareRoot(fhyperelliptic , x , y , t) = [ t2+1√−t6−t4−t2−1
, (t2+1)t√−t6−t4−t2−1
]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have theparametrization. Simply compute the Weierstrass form of thecurve.
Example
fhyperelliptic := x4 + y4 + x2 + y2;genus(fhyperelliptic , x , y) = 2;WeierstrassForm(fhyperelliptic) = y2 + x6 + x2 + x4 + 1
squareRoot(fhyperelliptic , x , y , t) = [ t2+1√−t6−t4−t2−1
, (t2+1)t√−t6−t4−t2−1
]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have theparametrization. Simply compute the Weierstrass form of thecurve.
Example
fhyperelliptic := x4 + y4 + x2 + y2;genus(fhyperelliptic , x , y) = 2;WeierstrassForm(fhyperelliptic) = y2 + x6 + x2 + x4 + 1
squareRoot(fhyperelliptic , x , y , t) = [ t2+1√−t6−t4−t2−1
, (t2+1)t√−t6−t4−t2−1
]
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 whichis not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there doesnot exist a square-root parametrization.
Example
fnon−hyperelliptic := x3 − y5 + y4 − y2;genus(fnon−hyperelliptic , x , y) = 3;squareRoot(fnon−hyperelliptic , x , y , t);Error, It is not square-root parameterisable.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 whichis not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there doesnot exist a square-root parametrization.
Example
fnon−hyperelliptic := x3 − y5 + y4 − y2;genus(fnon−hyperelliptic , x , y) = 3;squareRoot(fnon−hyperelliptic , x , y , t);Error, It is not square-root parameterisable.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 whichis not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there doesnot exist a square-root parametrization.
Example
fnon−hyperelliptic := x3 − y5 + y4 − y2;
genus(fnon−hyperelliptic , x , y) = 3;squareRoot(fnon−hyperelliptic , x , y , t);Error, It is not square-root parameterisable.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 whichis not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there doesnot exist a square-root parametrization.
Example
fnon−hyperelliptic := x3 − y5 + y4 − y2;genus(fnon−hyperelliptic , x , y) = 3;
squareRoot(fnon−hyperelliptic , x , y , t);Error, It is not square-root parameterisable.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Definition and ProblemClassification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 whichis not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there doesnot exist a square-root parametrization.
Example
fnon−hyperelliptic := x3 − y5 + y4 − y2;genus(fnon−hyperelliptic , x , y) = 3;squareRoot(fnon−hyperelliptic , x , y , t);Error, It is not square-root parameterisable.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Further DirectionsReferences
Further Directions
I Study n-root parametrizations.
I Devise and implement algorithms for computing n-rootparametrizations.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Further DirectionsReferences
Further Directions
I Study n-root parametrizations.
I Devise and implement algorithms for computing n-rootparametrizations.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Further DirectionsReferences
Further Directions
I Study n-root parametrizations.
I Devise and implement algorithms for computing n-rootparametrizations.
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations
IntroductionSquare-root Parametrization
Further Directions
Further DirectionsReferences
References
D. Eisenbud, Commutative Algebra with a view toward AlgebraicGeometry, Graduate Texts in Mathematics 150, Springer BerlinHeidelberg New York, 1995.
R. Hartshorne, Algebraic Geometry, Graduate Texts inMathematics,Springer Berlin Heidelberg New York, 1977
I.R. Shafarevich, Basic Algebraic Geometry ,Springer BerlinHeidelberg New York, 1974
Jose Manuel Garcia Vallinas / Josef Schicho Square-Root Parametrizations