Projective quadric

In projective analytic geometry, a projective quadric is the set of zeros of a nontrivial, homogeneous, quadratic function in variables , which is understood as the coordinate representation of a set of points in the - dimensional projective space over a body . ${\ displaystyle q}$ ${\ displaystyle n + 1}$ ${\ displaystyle (x_ {0}, x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle n}$ ${\ displaystyle KP ^ {n}}$ ${\ displaystyle K}$

Projective quadrics can be represented by a symmetrical matrix , provided the characteristic of the body is not 2 . If this matrix can be diagonalized by an orthogonal matrix , then the equation that describes the quadric can be applied to a form by choosing a suitable projective coordinate system

{\ displaystyle \ alpha _ {0} x_ {0} ^ {2} + \ alpha _ {1} x_ {1} ^ {2} + \ cdots + \ alpha _ {r} x_ {r} ^ {2} = 0, \; \ alpha _ {j} \ neq 0}

to be brought. The number is the rank of the representation matrix , the coefficients are its eigenvalues different from 0 . Here, a different coefficient of 0 to the equation always by selecting an appropriate coordinate system with an arbitrary to him square equivalent coefficient to be converted, all the coefficients are only up to a common factor determined. The order of the coefficients can be chosen arbitrarily by a suitable basic transformation. ${\ displaystyle 1 \ leq r \ leq n + 1}$ ${\ displaystyle \ alpha _ {j}}$ ${\ displaystyle \ beta _ {j} = c ^ {2} \ cdot \ alpha _ {j}, c \ in K ^ {*} = K \ setminus \ lbrace 0 \ rbrace}$ ${\ displaystyle t \ in K ^ {*}}$

In synthetic geometry , quadrics in projective geometries are defined as point sets without coordinates. This makes it possible to investigate such point sets in non-Desarguean planes and non-Pappusian spaces . → See also Quadratic Set .

Definitions

Homogeneous quadratic polynomial

Be a body . A polynomial in (at most) variables is called a homogeneous quadratic polynomial if it is a sum of quadratic monomials of the form . Such a polynomial leaves a representation ${\ displaystyle K}$ ${\ displaystyle n + 1}$ ${\ displaystyle p \ in K [x_ {0}, x_ {1}, \ ldots, x_ {n}]}$ ${\ displaystyle r_ {jk} \ cdot x_ {j} \ cdot x_ {k} (r_ {jk} \ in K; 0 \ leq j \ leq k \ leq n)}$

{\ displaystyle p (x_ {0}, x_ {1}, \ ldots, x_ {n}) = {\ vec {x}} ^ {T} \ cdot A \ cdot {\ vec {x}}}

to, where is a square matrix. If the matrix is required to be symmetrical, then it is uniquely determined by the coefficients of the monomials. It then applies ${\ displaystyle A}$ ${\ displaystyle (n + 1) \ times (n + 1)}$ ${\ displaystyle A = (a_ {jk}) _ {(0 \ leq j, k \ leq n)}}$ ${\ displaystyle r_ {jk}}$

{\ displaystyle a_ {jk} = {\ begin {cases} r_ {jk}, \ quad {\ text {falls}} \, j = k \\ {\ frac {1} {2}} \ cdot r_ {jk }, \; {\ text {falls}} \, j \ neq k, \ end {cases}}}

if the characteristic of the body is not 2. For bodies with characteristic 2, it is generally not possible to represent them using a symmetrical matrix. ${\ displaystyle K}$

Homogeneous quadratic function

Every homogeneous quadratic polynomial becomes a homogeneous, quadratic function

{\ displaystyle q: K ^ {n + 1} \ rightarrow K; \ quad {\ begin {pmatrix} x_ {0} \\\ vdots \\ x_ {n} \ end {pmatrix}} \ mapsto p (x_ { 0}, \ ldots, x_ {n}) = (x_ {0}, \ ldots, x_ {n}) \ cdot A \ cdot {\ begin {pmatrix} x_ {0} \\\ vdots \\ x_ {n } \ end {pmatrix}}}

defined on the vector space . Is the zero polynomial and thus the symmetric matrix the zero matrix , then the quadratic function is trivial , in all other cases, ie when the rank of greater than or equal to 1, is not trivial . ${\ displaystyle K ^ {n + 1}}$ ${\ displaystyle p}$ ${\ displaystyle A}$ ${\ displaystyle A}$

Projective quadric

For a homogeneous, quadratic function , if and only if applies for every scalar . If one therefore selects a fixed projective coordinate system in a -dimensional projective space and thus identifies the space with its standard model , then a set of points in the projective space is described. For the trivial quadratic function that is the entire space. In all other cases the satisfying set of the coordinate equation, i.e. the point set ${\ displaystyle q: K ^ {n + 1} \ rightarrow K}$ ${\ displaystyle q ({\ vec {x}}) = 0}$ ${\ displaystyle q (t \ cdot {\ vec {x}}) = 0}$ ${\ displaystyle t \ in K}$ ${\ displaystyle n}$ ${\ displaystyle K}$ ${\ displaystyle KP ^ {n} {\ mathrel {\ widehat {=}}} K ^ {n + 1}}$ ${\ displaystyle q ({\ vec {x}}) = 0}$ ${\ displaystyle q \ equiv 0}$

{\ displaystyle \ lbrace P = \ left [{\ vec {x}} \ right] \ in KP ^ {n} | q ({\ vec {x}}) = 0 \ rbrace}

referred to as the projective quadric .

Equivalence of Quadrics

Two quadrics, which are described by the nontrivial quadratic functions or , are called projectively equivalent if there are symmetrical representation matrices of and of , a regular matrix and a body element such that ${\ displaystyle q_ {1} ({\ vec {x}})}$ ${\ displaystyle q_ {2} ({\ vec {x}})}$ ${\ displaystyle A}$ ${\ displaystyle q_ {1}}$ ${\ displaystyle B}$ ${\ displaystyle q_ {2}}$ ${\ displaystyle S}$ ${\ displaystyle t \ in K ^ {*}}$

{\ displaystyle S ^ {T} \ times A \ times S = t \ times B}

holds ( is the transposed matrix ).

{\ displaystyle S ^ {T}}

It is thus possible, precisely for equivalent quadrics, to bring the equation of one quadric to the form of the second or a multiple of this form by choosing a suitable projective coordinate system. For the equivalence it is necessary that the quadrics can be mapped bijectively onto one another as point sets through a projectivity , which can then be represented by . This is also sufficient for projective spaces over an algebraically closed field . ${\ displaystyle S}$

A quadric that has a symmetrical representation matrix with full rank , i.e. of rank , is referred to as non-degenerate , every other quadric as degenerate . A quadric whose fulfillment set holds true, i.e. which does not contain a projective point, is called a zero-part projective quadric . ${\ displaystyle n + 1}$ ${\ displaystyle \ lbrace {\ vec {x}} \ in K ^ {n + 1} | q ({\ vec {x}}) = 0 \ rbrace = \ lbrace 0 \ rbrace}$

Degenerate quadrics are never zero-part.

Invariants

The rank of a symmetric representation matrix is a projective invariant for each quadric , it is also referred to as the rank of the projective quadric. ${\ displaystyle r}$ ${\ displaystyle A}$ ${\ displaystyle q ({\ vec {x}}) = 0}$

For the case of a projective quadric over a real projective space, Sylvester's theorem of inertia provides an invariant: Since every eigenvalue of the symmetrical representation matrix is 0 or is quadratically equivalent to +1 or −1, one can achieve - possibly by multiplying by −1 that the number of positive eigenvalues is not less than the number of negative eigenvalues. The number is another projective invariant for real quadrics, which is sometimes referred to as the projective signature , since, together with the rank, it contains the essential information of the signature of a bilinear form for projective quadrics . Two real quadrics are equivalent if and only if they agree in their rank and their projective signature . A normal form can always be selected in which and therefore is. ${\ displaystyle r _ {+}}$ ${\ displaystyle r _ {-}}$ ${\ displaystyle s: = \ min \ lbrace r _ {+}, r _ {-} \ rbrace}$ ${\ displaystyle r}$ ${\ displaystyle s}$ ${\ displaystyle s = r _ {-}}$ ${\ displaystyle 0 \ leq r _ {-} \ leq r / 2}$

Projective and affine classification of quadrics

By slitting a projective space ( i.e. by selecting a hyperplane of the room as a remote hyperplane ), an (affine) quadric arises from each projective quadric in the affine space created in the process . Usually two different affine quadrics are only counted of the same type if the associated projective quadrics in the projective closure of the affine space belong to the same projective type. Therefore, the affine typing, especially for zero-part quadrics, only becomes complete through typing in the projective closure.

Examples

The ordinary cases of linear algebra

In the following cases, the symmetrical representation matrix can always be diagonalized orthogonally:

In a complex projective space of the dimension there is exactly one type of a projective quadric with a representation matrix from the rank to each rank , so in total exactly different types. No quadric is zero-part. ${\ displaystyle n}$ ${\ displaystyle r}$ ${\ displaystyle 1 \ leq r \ leq n + 1}$ ${\ displaystyle n + 1}$

The same applies to every algebraically closed field with characteristics different from 2.

In a real projective space of dimension there are exactly different types of projective quadrics for every possible rank with the representation matrix. Where is the largest whole number , compare Gaussian brackets . ${\ displaystyle n}$ ${\ displaystyle r}$ ${\ displaystyle 1 \ leq r \ leq n + 1}$ ${\ displaystyle \ lfloor {\ frac {r} {2}} \ rfloor +1}$ ${\ displaystyle \ lfloor {\ frac {r} {2}} \ rfloor}$ ${\ displaystyle z \ leq {\ frac {r} {2}}}$

The same applies to every real closed body .

In a projective plane , i.e. a two- dimensional projective space over a Euclidean body , for example the real numbers, there are exactly 5 different types of projective quadrics, two of which have representation matrices with the full rank 3 and the normal form or . The first of the normal forms mentioned describes a zero-part quadric. The normal forms of the degenerate quadrics are . ${\ displaystyle x_ {0} ^ {2} + x_ {1} ^ {2} + x_ {2} ^ {2} = 0}$ ${\ displaystyle x_ {0} ^ {2} + x_ {1} ^ {2} -x_ {2} ^ {2} = 0}$ ${\ displaystyle x_ {0} ^ {2} = 0, x_ {0} ^ {2} + x_ {1} ^ {2} = 0, x_ {0} ^ {2} -x_ {1} ^ {2 } = 0}$
In a three- dimensional projective space over a real closed body, for example the real numbers, there are exactly 8 different types of projective quadrics. Three of them have full rank and the normal form , or . The first of the mentioned normal forms describes a zero-part quadric, the normal forms of the degenerate quadrics are described by the 5 normal forms of the two-dimensional case. ${\ displaystyle x_ {0} ^ {2} + x_ {1} ^ {2} + x_ {2} ^ {2} + x_ {3} ^ {2} = 0}$ ${\ displaystyle x_ {0} ^ {2} + x_ {1} ^ {2} + x_ {2} ^ {2} -x_ {3} ^ {2} = 0}$ ${\ displaystyle x_ {0} ^ {2} + x_ {1} ^ {2} -x_ {2} ^ {2} -x_ {3} ^ {2} = 0}$

Practical calculation of a rational normal form

For practical calculations, the diagonalization of the symmetrical matrix - even with computer algebra systems - is time-consuming and initially only leads to real and complex normal forms. Instead, for practical calculations, the mixed terms of the quadratic function will be eliminated by completing the square . This should be made clear here using a numerical example:

The quadratic function is given by the symmetric matrix ${\ displaystyle q_ {A}}$

{\ displaystyle A = {\ begin {pmatrix} 1 & 2 & 3 \\ 2 & 2 & -1 \\ 3 & -1 & 1 \ end {pmatrix}}}

is represented, so

{\ displaystyle q_ {A} ({\ vec {x}}) = x_ {0} ^ {2} + 4x_ {0} x_ {1} + 6x_ {0} x_ {2} + 2x_ {1} ^ { 2} -2x_ {1} x_ {2} + x_ {2} ^ {2}.}

Now add three squares:

${\ displaystyle x_ {0} = y_ {0} -2x_ {1}}$ leads to the mold , ${\ displaystyle y_ {0} ^ {2} -2x_ {1} ^ {2} + 6y_ {0} x_ {2} -14x_ {1} x_ {2} + x_ {2} ^ {2}}$
${\ displaystyle y_ {0} = z_ {0} -3x_ {2}}$ then carries on , ${\ displaystyle z_ {0} ^ {2} -2x_ {1} ^ {2} -14x_ {1} x_ {2} -8x_ {2} ^ {2}}$
${\ displaystyle x_ {1} = y_ {1} - {\ frac {7} {2}} x_ {2}}$ performs on . ${\ displaystyle z_ {0} ^ {2} -2y_ {1} ^ {2} + {\ frac {33} {2}} x_ {2} ^ {2}}$

The transition matrices are included

{\ displaystyle S_ {1} = {\ begin {pmatrix} 1 & -2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \ end {pmatrix}}, S_ {2} = {\ begin {pmatrix} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \ end {pmatrix}}, S_ {3} = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & - {\ frac {7} {2}} \\ 0 & 0 & 1 \ end {pmatrix}},}

whose product , together with the factor, gives the equivalence of the matrix to the diagonal matrix , because it is . This means that the quadric described by is describable over every body whose characteristic is not 2 . About reads a normal form , because there −4 and −1 are square equivalent. If the quadric lies in one plane, then it is degenerate if and only if the characteristic of the body is 2, 3 or 11, for all other characteristics its rank is 3. Above the real numbers, its rank is 3 and its projective signature is 1. ${\ displaystyle S = S_ {1} \ cdot S_ {2} \ cdot S_ {3}}$ ${\ displaystyle t = 1/2}$ ${\ displaystyle A}$ ${\ displaystyle D = \ operatorname {diag} (2, -4,33)}$ ${\ displaystyle S ^ {T} \ cdot A \ cdot S = t \ cdot D}$ ${\ displaystyle A}$ ${\ displaystyle 2x_ {0} ^ {2} -4x_ {1} ^ {2} + 33x_ {2} ^ {2}}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle 2x_ {0} ^ {2} + 33x_ {1} ^ {2} -x_ {2} ^ {2}}$

literature

The first three writings by Helmut Hasse are fundamental for the theory of the quadratic forms and quadrics over the rational numbers , the fourth extends this to number fields :

Helmut Hasse: About the representability through square forms in the body of rational numbers . In: Journal for pure and applied mathematics . tape 152 , 1923, pp. 129–148 ( digizeitschriften.de ).
Helmut Hasse: About the equivalence of quadratic forms in the field of rational numbers . In: Journal for pure and applied mathematics . tape 152 , 1923, pp. 205–224 ( digizeitschriften.de ).
Helmut Hasse: Symmetrical matrices in the field of rational numbers . In: Journal for pure and applied mathematics . tape 153 , 1924, pp. 12-43 ( digizeitschriften.de ).
Helmut Hasse: Representability of numbers by square forms in any algebraic number field . In: Journal for pure and applied mathematics . tape 153 , 1924, pp. 113-130 ( digizeitschriften.de ).

Leutbecher's textbook brings some applications of the quadrics to Diophantine equations . Such applications can be found in most introductory books on algebraic number theory:

Armin Leutbecher: Number Theory. An introduction to algebra . Springer, Berlin a. a. 1996, ISBN 3-540-58791-8 .

Schaal's textbook provides the classification of complex and real quadrics in the projective as well as in the affine and for real in the Euclidean sense and also shows the relationships between these classifications for the two- and three-dimensional case. Similar things can be found in many textbooks on linear algebra:

Hermann Schaal: Linear Algebra and Analytical Geometry. Volume 1-2. Vieweg, Braunschweig 1976, ISBN 3-528-03056-9 (Vol. 1), ISBN 3-528-03057-7 (Vol. 2), (Volume 2. 2nd, revised edition. Ibid 1980, ISBN 3-528 -13057-1 ).