Introduction
Square-root Parametrization
Further Directions
Square-Root Parametrizations
Jose Manuel Garcia Vallinas / Josef Schicho
Spezialforschungsbereich F013
Subproject F1303
Johann Radon Institute for Computional and Applied Mathematics (RICAM)
Austrian Academy of Sciences (ÖAW)
Linz, Austria
Workshop on Algebraic Spline Curves and Surfaces
May 17-18, 2006, Eger, Hungary
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definitions I
Definitions II
Weierstrass Form
Definitions I
Plane Algebraic Curve
An affine irreducible plane algebraic curve over C is defined as
the set
C = {(a, b) ∈ A2 (C)|f (a, b) = 0}
for a non-constant irreducible polynomial f (x, y ) ∈ C[x, y ].
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definitions I
Definitions II
Weierstrass Form
Definitions I
Plane Algebraic Curve
An affine irreducible plane algebraic curve over C is defined as
the 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 the
order of the first non-vanishing term in the Taylor expansion of f
at P is greater than 1.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definitions I
Definitions II
Weierstrass Form
Definitions II
Genus
Let C be an irreducible plane curve of degree n, having only
ordinary singularities of multiplicities r1 , . . . , rm . The genus of C ,
g (C ) , is defined as
m
X
1
ri (ri − 1)]
g (C ) := [(n − 1)(n − 2) −
2
i=1
For non-ordinary singularities, the genus can be computed similarly.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definitions I
Definitions II
Weierstrass Form
Weierstrass Form
A curve C is called elliptic if and only if its genus is 1
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definitions I
Definitions II
Weierstrass 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 to
the form y 2 = F (x), where F (x) is a square-free polynomial in x
of degree 3.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definitions I
Definitions II
Weierstrass 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 to
the form y 2 = F (x), where F (x) is a square-free polynomial in x
of degree 3.
A curve C is called hyper-elliptic if and only if its genus g is
greater than 1 and it can be birationally transformed to y 2 = 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
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Square-root Parametrization Problem
Square-root Parameterisable Curve
A curve is called square-root parameterisable
if it can be
p
parameterised in terms of t and P(t), where P(t) is a
polynomial in t.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Square-root Parametrization Problem
Square-root Parameterisable Curve
A curve is called square-root parameterisable
if it can be
p
parameterised in terms of t and P(t), where P(t) is a
polynomial in t.
Example: x 2 + y 2 − 1:
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Square-root Parametrization Problem
Square-root Parameterisable Curve
A curve is called square-root parameterisable
if it can be
p
parameterised in terms of t and P(t), where P(t) is a
polynomial in t.
√
√
Example: x 2 + y 2 − 1: t 7→ (t, 1 − t 2 ) or ( 1 − t 2 , t)
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Square-root Parametrization Problem
Square-root Parameterisable Curve
A curve is called square-root parameterisable
if it can be
p
parameterised in terms of t and P(t), where P(t) is a
polynomial in t.
√
√
Example: x 2 + y 2 − 1: t 7→ (t, 1 − t 2 ) or ( 1 − t 2 , t)
Problem
We have an algebraic (irreducible) plane curve C and we want to
know if it is square-root parameterisable and compute this
parametrization in the positive case.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Rational Curves
If we have a rational curve, we take its rational parametrization.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Rational Curves
If we have a rational curve, we take its rational parametrization.
Example
frational := y 2 − x 2 (x + 1);
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Rational Curves
If we have a rational curve, we take its rational parametrization.
Example
frational := y 2 − x 2 (x + 1);
genus(frational , x, y ) = 0;
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Rational Curves
If we have a rational curve, we take its rational parametrization.
Example
frational := y 2 − x 2 (x + 1);
genus(frational , x, y ) = 0;
squareRoot(frational , x, y , t) = [−1 + t 2 , t(−1 + t 2 )]
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Elliptic Curves
If we have an elliptic curve, we simply need to compute its
Weierstrass form and substitute.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Elliptic Curves
If we have an elliptic curve, we simply need to compute its
Weierstrass form and substitute.
Example
felliptic := x 3 − y 3 − x;
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Elliptic Curves
If we have an elliptic curve, we simply need to compute its
Weierstrass form and substitute.
Example
felliptic := x 3 − y 3 − x;
genus(felliptic , x, y ) = 1;
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Elliptic Curves
If we have an elliptic curve, we simply need to compute its
Weierstrass form and substitute.
Example
felliptic := x 3 − y 3 − x;
genus(felliptic , x, y ) = 1;
WeierstrassForm(felliptic ) = x 3 + y 2 − 1
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Elliptic Curves
If we have an elliptic curve, we simply need to compute its
Weierstrass form and substitute.
Example
felliptic := x 3 − y 3 − x;
genus(felliptic , x, y ) = 1;
WeierstrassForm(felliptic ) = x 3 √
+ y2 − 1
squareRoot(felliptic , x, y , t) = [
Jose Manuel Garcia Vallinas / Josef Schicho
−t 3 +1(t+1)
,
3+3t+3t 2
√
−t 3 +1(t+2)
]
3+3t+3t 2
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have the
parametrization. Simply compute the Weierstrass form of the
curve.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have the
parametrization. Simply compute the Weierstrass form of the
curve.
Example
fhyperelliptic := x 4 + y 4 + x 2 + y 2 ;
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have the
parametrization. Simply compute the Weierstrass form of the
curve.
Example
fhyperelliptic := x 4 + y 4 + x 2 + y 2 ;
genus(fhyperelliptic , x, y ) = 2;
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have the
parametrization. Simply compute the Weierstrass form of the
curve.
Example
fhyperelliptic := x 4 + y 4 + x 2 + y 2 ;
genus(fhyperelliptic , x, y ) = 2;
WeierstrassForm(fhyperelliptic ) = y 2 + x 6 + x 2 + x 4 + 1
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Hyper-elliptic Curves
In the case of a hyper-elliptic curve, we already have the
parametrization. Simply compute the Weierstrass form of the
curve.
Example
fhyperelliptic := x 4 + y 4 + x 2 + y 2 ;
genus(fhyperelliptic , x, y ) = 2;
WeierstrassForm(fhyperelliptic ) = y 2 + x 6 + x 2 + x 4 + 1
2
2
squareRoot(fhyperelliptic , x, y , t) = [ √−t 6t−t+1
, √−t(t6 −t+1)t
]
4 −t 2 −1
4 −t 2 −1
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 which
is not hyperelliptic.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 which
is not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there does
not exist a square-root parametrization.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 which
is not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there does
not exist a square-root parametrization.
Example
fnon−hyperelliptic := x 3 − y 5 + y 4 − y 2 ;
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 which
is not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there does
not exist a square-root parametrization.
Example
fnon−hyperelliptic := x 3 − y 5 + y 4 − y 2 ;
genus(fnon−hyperelliptic , x, y ) = 3;
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Definition and Problem
Classification
Non-hyperelliptic Curves
A non-hyperelliptic curve is a curve of genus greater than 2 which
is not hyperelliptic.
If the curve is non-hyperelliptic, it can be shown that there does
not exist a square-root parametrization.
Example
fnon−hyperelliptic := x 3 − y 5 + y 4 − y 2 ;
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
Introduction
Square-root Parametrization
Further Directions
Further Directions
References
Further Directions
I
Study n-root parametrizations.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Further Directions
References
Further Directions
I
Study n-root parametrizations.
I
Devise and implement algorithms for computing n-root
parametrizations.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Further Directions
References
Further Directions
I
Study n-root parametrizations.
I
Devise and implement algorithms for computing n-root
parametrizations.
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations
Introduction
Square-root Parametrization
Further Directions
Further Directions
References
References
D. Eisenbud, Commutative Algebra with a view toward Algebraic
Geometry, Graduate Texts in Mathematics 150, Springer Berlin
Heidelberg New York, 1995.
R. Hartshorne, Algebraic Geometry, Graduate Texts in
Mathematics,Springer Berlin Heidelberg New York, 1977
I.R. Shafarevich, Basic Algebraic Geometry ,Springer Berlin
Heidelberg New York, 1974
Jose Manuel Garcia Vallinas / Josef Schicho
Square-Root Parametrizations