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Homogeneous polynomial of even degree in two variables
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Concerning this paper in which $k(n,d)$ is the set of all homogeneous polynomials in $n$ variables of even degree $d$ over a field $k$.
Page 281 says:
"Let us denote $nabla(n, d, mathbbC) subset mathbbC(n,d)$ the set of singular polynomials of degree $d$ in $n$ variables, over the complex numbers. That is, $nabla(n, d, mathbbC) = textthere exists x in mathbbC^n-0, fracpartial Fpartial x_i(x)=0, forall i$.
It is known that $nabla(n, d, mathbbC)$ is an irreducible algebraic hypersurface of degree $D = n(d − 1)^n−1$
defined over the rational numbers. Therefore, there exists a polynomial (unique up to multiplicative constant)
$Delta = Delta(n, d)$ called the discriminant, such that
$nabla(n, d, mathbbC) = F in mathbbC(n, d) $".
For simplicity, let us concentrate on the case $n=2$.
(1) Could one please explain or give a reference to "Therefore"?
(2) More important (to me, at the moment), given a homogeneous polynomial of degree $d$ in two variables $x,y$ over $mathbbR$, how to find $Delta$? In this case, its degree $D$ should be $2(d-1)$.
Thank you very much!
algebraic-geometry polynomials real-algebraic-geometry
asked Jul 15 at 19:30
user237522
1,8141617
add a comment |Â
up vote
0
down vote
favorite
Concerning this paper in which $k(n,d)$ is the set of all homogeneous polynomials in $n$ variables of even degree $d$ over a field $k$.
Page 281 says:
"Let us denote $nabla(n, d, mathbbC) subset mathbbC(n,d)$ the set of singular polynomials of degree $d$ in $n$ variables, over the complex numbers. That is, $nabla(n, d, mathbbC) = textthere exists x in mathbbC^n-0, fracpartial Fpartial x_i(x)=0, forall i$.
It is known that $nabla(n, d, mathbbC)$ is an irreducible algebraic hypersurface of degree $D = n(d − 1)^n−1$
defined over the rational numbers. Therefore, there exists a polynomial (unique up to multiplicative constant)
$Delta = Delta(n, d)$ called the discriminant, such that
$nabla(n, d, mathbbC) = F in mathbbC(n, d) $".
For simplicity, let us concentrate on the case $n=2$.
(1) Could one please explain or give a reference to "Therefore"?
(2) More important (to me, at the moment), given a homogeneous polynomial of degree $d$ in two variables $x,y$ over $mathbbR$, how to find $Delta$? In this case, its degree $D$ should be $2(d-1)$.
Thank you very much!
algebraic-geometry polynomials real-algebraic-geometry
asked Jul 15 at 19:30
user237522
1,8141617
1
(1) Hypersurfaces are, basically by definition, given by the vanishing of a single equation; polynomial since algebraic. (2) $Delta$ is not associated to a single polynomial, but to the set of all polynomials of fixed degree and number of variables. For example, consider conics $ax^2 + by^2 + cxy + dx + ey + f = 0$. Then there is a vector space isomorphic to $mathbb R^6$ in which the point $(a,b,c,d,e,f)$ parametrizes a given conic. The discriminant $Delta(2,2)$ is a cubic polynomial in $a,b,c,d,e,f$ given by a certain determinant. Check en.wikipedia.org/wiki/Discriminant
– Tabes Bridges
Jul 15 at 21:15
Thank you! Could you please explain what is the sufficient condition (= the condition that guarantees that $f$ is non-negative) of Proposition 2.9 for $f=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$, $a,b,c,d,e in mathbbR$?
– user237522
Jul 15 at 21:27
Perhaps I should first concentrate on the case $d=2$. Denote: $f=ax^2+bxy+cy^2$. By Sylvester's theorem $f$ is positive iff $a>0$ and $ac-fracb2fracb2 > 0$. I will try to apply Proposition 2.9 and see if I get the same condition (one of the two directions of the iff): $Delta(ax^2+bxy+cy^2+t(x^2+y^2))=-4t^2-4(a+c)t+b^2-4ac$
– user237522
Jul 15 at 22:20
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Concerning this paper in which $k(n,d)$ is the set of all homogeneous polynomials in $n$ variables of even degree $d$ over a field $k$.
Page 281 says:
"Let us denote $nabla(n, d, mathbbC) subset mathbbC(n,d)$ the set of singular polynomials of degree $d$ in $n$ variables, over the complex numbers. That is, $nabla(n, d, mathbbC) = textthere exists x in mathbbC^n-0, fracpartial Fpartial x_i(x)=0, forall i$.
It is known that $nabla(n, d, mathbbC)$ is an irreducible algebraic hypersurface of degree $D = n(d − 1)^n−1$
defined over the rational numbers. Therefore, there exists a polynomial (unique up to multiplicative constant)
$Delta = Delta(n, d)$ called the discriminant, such that
$nabla(n, d, mathbbC) = F in mathbbC(n, d) $".
For simplicity, let us concentrate on the case $n=2$.
(1) Could one please explain or give a reference to "Therefore"?
(2) More important (to me, at the moment), given a homogeneous polynomial of degree $d$ in two variables $x,y$ over $mathbbR$, how to find $Delta$? In this case, its degree $D$ should be $2(d-1)$.
Thank you very much!
algebraic-geometry polynomials real-algebraic-geometry
asked Jul 15 at 19:30
user237522
1,8141617
Concerning this paper in which $k(n,d)$ is the set of all homogeneous polynomials in $n$ variables of even degree $d$ over a field $k$.
Page 281 says:
"Let us denote $nabla(n, d, mathbbC) subset mathbbC(n,d)$ the set of singular polynomials of degree $d$ in $n$ variables, over the complex numbers. That is, $nabla(n, d, mathbbC) = textthere exists x in mathbbC^n-0, fracpartial Fpartial x_i(x)=0, forall i$.
It is known that $nabla(n, d, mathbbC)$ is an irreducible algebraic hypersurface of degree $D = n(d − 1)^n−1$
defined over the rational numbers. Therefore, there exists a polynomial (unique up to multiplicative constant)
$Delta = Delta(n, d)$ called the discriminant, such that
$nabla(n, d, mathbbC) = F in mathbbC(n, d) $".
For simplicity, let us concentrate on the case $n=2$.
(1) Could one please explain or give a reference to "Therefore"?
(2) More important (to me, at the moment), given a homogeneous polynomial of degree $d$ in two variables $x,y$ over $mathbbR$, how to find $Delta$? In this case, its degree $D$ should be $2(d-1)$.
Thank you very much!
algebraic-geometry polynomials real-algebraic-geometry
asked Jul 15 at 19:30
user237522
1,8141617
edited Jul 15 at 20:09
asked Jul 15 at 19:30
user237522
1,8141617
asked Jul 15 at 19:30
user237522
1,8141617
asked Jul 15 at 19:30
user237522
1,8141617
1,8141617
1
(1) Hypersurfaces are, basically by definition, given by the vanishing of a single equation; polynomial since algebraic. (2) $Delta$ is not associated to a single polynomial, but to the set of all polynomials of fixed degree and number of variables. For example, consider conics $ax^2 + by^2 + cxy + dx + ey + f = 0$. Then there is a vector space isomorphic to $mathbb R^6$ in which the point $(a,b,c,d,e,f)$ parametrizes a given conic. The discriminant $Delta(2,2)$ is a cubic polynomial in $a,b,c,d,e,f$ given by a certain determinant. Check en.wikipedia.org/wiki/Discriminant
– Tabes Bridges
Jul 15 at 21:15
Thank you! Could you please explain what is the sufficient condition (= the condition that guarantees that $f$ is non-negative) of Proposition 2.9 for $f=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$, $a,b,c,d,e in mathbbR$?
– user237522
Jul 15 at 21:27
Perhaps I should first concentrate on the case $d=2$. Denote: $f=ax^2+bxy+cy^2$. By Sylvester's theorem $f$ is positive iff $a>0$ and $ac-fracb2fracb2 > 0$. I will try to apply Proposition 2.9 and see if I get the same condition (one of the two directions of the iff): $Delta(ax^2+bxy+cy^2+t(x^2+y^2))=-4t^2-4(a+c)t+b^2-4ac$
– user237522
Jul 15 at 22:20
add a comment |Â
1
(1) Hypersurfaces are, basically by definition, given by the vanishing of a single equation; polynomial since algebraic. (2) $Delta$ is not associated to a single polynomial, but to the set of all polynomials of fixed degree and number of variables. For example, consider conics $ax^2 + by^2 + cxy + dx + ey + f = 0$. Then there is a vector space isomorphic to $mathbb R^6$ in which the point $(a,b,c,d,e,f)$ parametrizes a given conic. The discriminant $Delta(2,2)$ is a cubic polynomial in $a,b,c,d,e,f$ given by a certain determinant. Check en.wikipedia.org/wiki/Discriminant
– Tabes Bridges
Jul 15 at 21:15
Thank you! Could you please explain what is the sufficient condition (= the condition that guarantees that $f$ is non-negative) of Proposition 2.9 for $f=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$, $a,b,c,d,e in mathbbR$?
– user237522
Jul 15 at 21:27
Perhaps I should first concentrate on the case $d=2$. Denote: $f=ax^2+bxy+cy^2$. By Sylvester's theorem $f$ is positive iff $a>0$ and $ac-fracb2fracb2 > 0$. I will try to apply Proposition 2.9 and see if I get the same condition (one of the two directions of the iff): $Delta(ax^2+bxy+cy^2+t(x^2+y^2))=-4t^2-4(a+c)t+b^2-4ac$
– user237522
Jul 15 at 22:20
1
1
(1) Hypersurfaces are, basically by definition, given by the vanishing of a single equation; polynomial since algebraic. (2) $Delta$ is not associated to a single polynomial, but to the set of all polynomials of fixed degree and number of variables. For example, consider conics $ax^2 + by^2 + cxy + dx + ey + f = 0$. Then there is a vector space isomorphic to $mathbb R^6$ in which the point $(a,b,c,d,e,f)$ parametrizes a given conic. The discriminant $Delta(2,2)$ is a cubic polynomial in $a,b,c,d,e,f$ given by a certain determinant. Check en.wikipedia.org/wiki/Discriminant
– Tabes Bridges
Jul 15 at 21:15
(1) Hypersurfaces are, basically by definition, given by the vanishing of a single equation; polynomial since algebraic. (2) $Delta$ is not associated to a single polynomial, but to the set of all polynomials of fixed degree and number of variables. For example, consider conics $ax^2 + by^2 + cxy + dx + ey + f = 0$. Then there is a vector space isomorphic to $mathbb R^6$ in which the point $(a,b,c,d,e,f)$ parametrizes a given conic. The discriminant $Delta(2,2)$ is a cubic polynomial in $a,b,c,d,e,f$ given by a certain determinant. Check en.wikipedia.org/wiki/Discriminant
– Tabes Bridges
Jul 15 at 21:15
Thank you! Could you please explain what is the sufficient condition (= the condition that guarantees that $f$ is non-negative) of Proposition 2.9 for $f=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$, $a,b,c,d,e in mathbbR$?
– user237522
Jul 15 at 21:27
Thank you! Could you please explain what is the sufficient condition (= the condition that guarantees that $f$ is non-negative) of Proposition 2.9 for $f=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$, $a,b,c,d,e in mathbbR$?
– user237522
Jul 15 at 21:27
Perhaps I should first concentrate on the case $d=2$. Denote: $f=ax^2+bxy+cy^2$. By Sylvester's theorem $f$ is positive iff $a>0$ and $ac-fracb2fracb2 > 0$. I will try to apply Proposition 2.9 and see if I get the same condition (one of the two directions of the iff): $Delta(ax^2+bxy+cy^2+t(x^2+y^2))=-4t^2-4(a+c)t+b^2-4ac$
– user237522
Jul 15 at 22:20
Perhaps I should first concentrate on the case $d=2$. Denote: $f=ax^2+bxy+cy^2$. By Sylvester's theorem $f$ is positive iff $a>0$ and $ac-fracb2fracb2 > 0$. I will try to apply Proposition 2.9 and see if I get the same condition (one of the two directions of the iff): $Delta(ax^2+bxy+cy^2+t(x^2+y^2))=-4t^2-4(a+c)t+b^2-4ac$
– user237522
Jul 15 at 22:20
add a comment |Â
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1
(1) Hypersurfaces are, basically by definition, given by the vanishing of a single equation; polynomial since algebraic. (2) $Delta$ is not associated to a single polynomial, but to the set of all polynomials of fixed degree and number of variables. For example, consider conics $ax^2 + by^2 + cxy + dx + ey + f = 0$. Then there is a vector space isomorphic to $mathbb R^6$ in which the point $(a,b,c,d,e,f)$ parametrizes a given conic. The discriminant $Delta(2,2)$ is a cubic polynomial in $a,b,c,d,e,f$ given by a certain determinant. Check en.wikipedia.org/wiki/Discriminant
– Tabes Bridges
Jul 15 at 21:15
Thank you! Could you please explain what is the sufficient condition (= the condition that guarantees that $f$ is non-negative) of Proposition 2.9 for $f=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$, $a,b,c,d,e in mathbbR$?
– user237522
Jul 15 at 21:27
Perhaps I should first concentrate on the case $d=2$. Denote: $f=ax^2+bxy+cy^2$. By Sylvester's theorem $f$ is positive iff $a>0$ and $ac-fracb2fracb2 > 0$. I will try to apply Proposition 2.9 and see if I get the same condition (one of the two directions of the iff): $Delta(ax^2+bxy+cy^2+t(x^2+y^2))=-4t^2-4(a+c)t+b^2-4ac$
– user237522
Jul 15 at 22:20