Annihilating Polynomials of Excellent Quadratic Forms Klaas-Tido R - PowerPoint PPT Presentation

Annihilating Polynomials of Excellent Quadratic Forms Klaas-Tido R¨ uhl EPFL GTEM Network - Number Fields, Lattices and Curves Cetraro, 02 Jun - 06 Jun 2008 Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 1 / 43

Introduction Introduction Already Witt knew that the Witt ring of a field is integral. But only in 1987 did David Lewis introduce specific annihilating polynomials. He proved that the polynomials P n := ( X − n )( X − n + 2) · · · ( X + n ) ∈ Z [ X ] , n ∈ N 0 , annihilate all n -dimensional quadratic forms over an arbitrary field. This initiated the study of annihilating polynomials of quadratic forms. Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 2 / 43

Quadratic Forms Quadratic Forms Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 3 / 43

Quadratic Forms Always N does not contain 0. We use N 0 := N ∪ { 0 } . We denote by K a field, char( K ) � = 2. Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 4 / 43

Quadratic Forms Quadratic Forms A quadratic space over K is a tuple ( V , b ) where V is an n -dimensional K -vector space, n ∈ N 0 , and b : V × V − → K is a symmetric K -bilinear form. An n -dimensional quadratic form over K , n ∈ N 0 , is a homogeneous element ϕ ∈ K [ X 1 , . . . , X n ] of degree 2. We write dim( ϕ ) = n . Note The dimension is an integral part of the definition of quadratic forms. For example X 2 1 ∈ K [ X 1 , X 2 , X 3 ] can be a quadratic form of dimension 3. Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 5 / 43

Quadratic Forms Quadratic Forms We can consider any n -dimensional quadratic form ϕ over K as a map ϕ : K n → K . The bilinear form associated to ϕ is defined as → 1 b ϕ : K n × K n − → K , ( v , w ) �− 2( ϕ ( v + w ) − ϕ ( v ) − ϕ ( w )) . The tuple ( K n , b ϕ ) is called quadratic space associated to ϕ . The matrix associated to ϕ is defined as A ϕ := ( b ϕ ( e i , e j )) i , j =1 ,..., n , where { e 1 , . . . , e n } is the standard basis of K n . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 6 / 43

Quadratic Forms Isometries Definition Two quadratic spaces ( V 1 , b 1 ) and ( V 2 , b 2 ) over K are called isometric if there exists a K -vector space isomorphism T : V 1 → V 2 such that b 1 ( v , w ) = b 2 ( Tv , Tw ) ∀ v , w ∈ V 1 . Definition Two quadratic forms ϕ and ψ over K are isometric if their associated quadratic spaces are isometric. We write ϕ ∼ = ψ . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 7 / 43

Quadratic Forms Diagonal Forms Let a 1 , . . . , a n ∈ K . We write � a 1 , . . . , a n � := a 1 X 2 1 + · · · + a n X 2 n ∈ K [ X 1 , . . . , X n ] . These forms are called diagonal forms. Theorem Let ϕ be an n -dimensional quadratic form over K . Then there exist a 1 , . . . , a n ∈ K such that ϕ ∼ = � a 1 , . . . , a n � . We are really only interested in quadratic forms up to isometry. Hence it suffices to consider only diagonal forms. Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 8 / 43

Quadratic Forms Operations There exists an orthogonal sum of two quadratic forms such that � a 1 , . . . , a n �⊥� b 1 , . . . , b m � = � a 1 , . . . , a n , b 1 , . . . , b m � . There exists a tensor product of two quadratic forms such that � a 1 , . . . , a n � ⊗ � b 1 , . . . , b m � ∼ = � a 1 b 1 , . . . , a 1 b m , a 2 b 1 , . . . , a n b m � . Note The analogous operations for quadratic spaces are defined as follows ( V 1 , b 1 ) ⊥ ( V 2 , b 2 ) := ( V 1 ⊕ V 2 , b 1 + b 2 ) and ( V 1 , b 1 ) ⊗ ( V 2 , b 2 ) := ( V 1 ⊗ K V 2 , b 1 b 2 ) . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 9 / 43

Quadratic Forms Non-degenerate Forms Definition A quadratic form ϕ is non-degenerate (or regular) if det( A ϕ ) � = 0. Henceforth we will only consider non-degenerate forms. Clear: ϕ ∼ ⇒ a 1 , . . . , a n ∈ K ∗ . = � a 1 , . . . , a n � is non-degenerate ⇐ Definition The determinant of a quadratic form ϕ ∼ = � a 1 , . . . , a n � is defined as det( ϕ ) := a 1 · · · a n ( K ∗ ) 2 ∈ K ∗ / ( K ∗ ) 2 . The determinant of a quadratic forms is well-defined, since we consider it in K ∗ / ( K ∗ ) 2 instead of in K ∗ . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 10 / 43

Quadratic Forms Isotropic Forms Definition A quadratic form ϕ over K is called isotropic if there exists 0 � = v ∈ K n such that ϕ ( v ) = 0. Otherwise ϕ is called anisotropic. By definition the zero-form over K is anisotropic. It is clear that every non-degenerate 1-dimensional form over K is anisotropic. Theorem Up to isometry there exists only one (non-degenerate) 2-dimensional isotropic form over K , i.e. the hyperbolic plane H := � 1 , − 1 � ∼ = � a , − a � for all a ∈ K ∗ . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 11 / 43

Quadratic Forms Hyperbolic Forms Let ϕ be a quadratic form over K . We use the notation m × ϕ := ϕ ⊥ . . . ⊥ ϕ . � �� � m -times Definition A quadratic form ϕ over K is called hyperbolic if there exists an m ∈ N 0 such that ϕ ∼ = m × H . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 12 / 43

Quadratic Forms Witt decomposition Theorem Let ϕ be a form over K . There exists a decomposition ϕ ∼ = ϕ an ⊥ ( i ( ϕ ) × H ) such that ϕ an is anisotropic and uniquely determined up to isometry and i ( ϕ ) is uniquely determined. Definition The form ϕ an is called anisotropic kernel of ϕ , and i ( ϕ ) is the Witt index of ϕ . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 13 / 43

Quadratic Forms The Witt-Grothendieck Ring The isometry class of a quadratic form ϕ over K will be denoted by [ ϕ ]. The isometry classes of quadratic forms over K form a semi-ring � W + ( K ) with addition [ ϕ ] + [ ψ ] := [ ϕ ⊥ ψ ] . and multiplication [ ϕ ] · [ ψ ] := [ ϕ ⊗ ψ ] . By applying the Grothendieck Construction for semi-groups to � W + ( K ) we obtain the Witt-Grothendieck Ring � W ( K ). Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 14 / 43

Quadratic Forms The Witt-Grothendieck Ring The elements of � W ( K ) are formal differences [ ϕ ] − [ ψ ] and there exists an up to isometry unique anisotropic form χ over K and a unique m ∈ N such that [ ϕ ] − [ ψ ] = [ χ ] ± [ m × H ] . We extend the notion of dimension to a ring homomorphism dim([ ϕ ] − [ ψ ]) := dim( ϕ ) − dim( ψ ) ∈ Z . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 15 / 43

Quadratic Forms The Witt Ring Denote by H the principal ideal of � W ( K ) generated by [ H ]. The Witt Ring of K is defined as W ( K ) := � W ( K ) / H . If ϕ is a quadratic form over K , then { ϕ } will denote its equivalence class in W ( K ). We write ψ ∼ ϕ if ψ ∈ { ϕ } . Note The elements of W ( K ) classify anisotropic quadratic forms over K . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 16 / 43

Quadratic Forms Example Let ϕ be a quadratic form over R . There exist r , s ∈ N 0 such that ϕ ∼ = r × � 1 � ⊥ s × �− 1 � . We have dim( ϕ ) = r + s and sign( ϕ ) = r − s , where sign( ϕ ) denotes the signature. Since [ �− 1 � ] 2 = 1, it follows that W ( R ) ∼ � = Z [( { 1 , − 1 } , · )] . Since ϕ is anisotropic if and only if r = 0 or s = 0, we obtain W ( R ) ∼ = Z . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 17 / 43

Annihilating Polynomials Annihilating Polynomials Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 18 / 43

Annihilating Polynomials Annihilating Polynomials Let R be a unitary, commutative ring, and let ι : Z → R be the canonical ring homomorphism. Definition A polynomial P = z n X n + · · · + z 1 X + z 0 ∈ Z [ X ] is called annihilating polynomial of x ∈ R , if P ( x ) := ι ( z n ) x n + · · · + ι ( z 1 ) x + ι ( z n ) = 0 ∈ R . Definition The annihilating ideal of x ∈ R is defined as Ann x := { P ∈ Z [ X ] | P ( x ) = 0 } ⊂ Z [ X ] . Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 19 / 43

Annihilating Polynomials Annihilating Polynomials In our case we have to consider the canonical ring homomorphisms → � ι 1 : Z − W ( K ) and ι 2 : Z − → W ( K ) defined for m ∈ N by m �− → [ m × � 1 � ] resp. m �− → { m × � 1 �} Usually we will simply write m for its image via ι 1 and ι 2 in � W ( K ) resp. W ( K ). Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 20 / 43