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 Pn := (X − n)(X − n + 2) · · · (X + n) ∈ Z[X], n ∈ N0, annihilate all n-dimensional quadratic forms over an arbitrary field. This initiated the study of annihilating polynomials of quadratic forms.

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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 N0 := 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 ∈ N0, and b : V × V − → K is a symmetric K-bilinear form. An n-dimensional quadratic form over K, n ∈ N0, is a homogeneous element ϕ ∈ K[X1, . . . , Xn] 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[X1, X2, X3] 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 bϕ : K n × K n − → K, (v, w) − → 1 2(ϕ(v + w) − ϕ(v) − ϕ(w)). The tuple (K n, bϕ) is called quadratic space associated to ϕ. The matrix associated to ϕ is defined as Aϕ := (bϕ(ei, ej))i,j=1,...,n, where {e1, . . . , en} 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 (V1, b1) and (V2, b2) over K are called isometric if there exists a K-vector space isomorphism T : V1 → V2 such that b1(v, w) = b2(Tv, Tw) ∀ v, w ∈ V1.

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 a1, . . . , an ∈ K. We write a1, . . . , an := a1X 2

1 + · · · + anX 2 n ∈ K[X1, . . . , Xn].

These forms are called diagonal forms.

Theorem

Let ϕ be an n-dimensional quadratic form over K. Then there exist a1, . . . , an ∈ K such that ϕ ∼ = a1, . . . , an. 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 a1, . . . , an⊥b1, . . . , bm = a1, . . . , an, b1, . . . , bm. There exists a tensor product of two quadratic forms such that a1, . . . , an ⊗ b1, . . . , bm ∼ = a1b1, . . . , a1bm, a2b1, . . . , anbm.

Note

The analogous operations for quadratic spaces are defined as follows (V1, b1)⊥(V2, b2) := (V1 ⊕ V2, b1 + b2) and (V1, b1) ⊗ (V2, b2) := (V1 ⊗K V2, b1b2).

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(Aϕ) = 0. Henceforth we will only consider non-degenerate forms. Clear: ϕ ∼ = a1, . . . , an is non-degenerate ⇐ ⇒ a1, . . . , an ∈ K ∗.

Definition

The determinant of a quadratic form ϕ ∼ = a1, . . . , an is defined as det(ϕ) := a1 · · · an(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 ∈ N0 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 ∼ = ϕan ⊥ (i(ϕ) × H) such that ϕan is anisotropic and uniquely determined up to isometry and i(ϕ) is uniquely determined.

Definition

The form ϕan 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

• btain 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 ∈ N0 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 = znX n + · · · + z1X + z0 ∈ Z[X] is called annihilating polynomial of x ∈ R, if P(x) := ι(zn)xn + · · · + ι(z1)x + ι(zn) = 0 ∈ R.

Definition

The annihilating ideal of x ∈ R is defined as Annx := {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 ι1 : Z − → W (K) and ι2 : 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 ι1 and ι2 in W (K) resp. W (K).

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 20 / 43

Annihilating Polynomials

Hurrelbrink’s method

Let G := K ∗/(K ∗)2 be the square class group of K. For a ∈ K ∗, denote by a its image in G. There exists a canonical ring homomorphism π1 : Z[G] − → W (K) defined by a1 + · · · + an − → [a1, . . . , an] . If π : W (K) → W (K) is the canonical projection, then we obtain a canonical homomorphism π2 := π ◦ π1 : Z[G] − → W (K).

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 21 / 43

Annihilating Polynomials

Hurrelbrink’s method

Let ϕ be a quadratic form over K, and let f ∈ π−1

1 ([ϕ]) ⊂ Z[G].

If P ∈ Z[X] is an annihilating polynomial of f . = ⇒ P is an annihilating polynomial of [ϕ]. = ⇒ P is an annihilating polynomial of {ϕ}. But Annf is easy to calculate.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 22 / 43

Annihilating Polynomials

Hurrelbrink’s method

Let Hom(Z[G], Z) be the set of ring homomorphisms Z[G] → Z. The set Sf := {χ(f ) | χ ∈ Hom(Z[G], Z)} is finite. Hence we can define Pf :=

• χ(f )∈Sf

(X − χ(f )). Then Annf = (Pf ) ⊂ Z[G].

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 23 / 43

Annihilating Polynomials

The Quadratic Form Case

In the case of quadratic forms, the situation is considerably more complicated. Let R be either W (K) or W (K), and let x ∈ R. Set Ssign

x

:= {χ(x) | χ ∈ Hom(R, Z)} and Qx :=

• χ(x)∈Ssign

x

(X − χ(x)).

Theorem

The greatest common divisor of the elements of Annx is equal to Qx. Furthermore Qx(x) ∈ R is a 2-torsion element.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 24 / 43

Annihilating Polynomials

The Quadratic Form Case

Definition

The polynomial Qx from the previous theorem is called embracing polynomial or signature polynomial. The name “embracing polynomial” was chosen in view of the fact, that we have Annx ⊂ (Qx), and (Qx) is the unique minimal principal ideal containing Annx The name “signature polynomial” stems from the fact, that Hom(W (K), Z) is just the set of signature homomorphisms.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 25 / 43

Annihilating Polynomials

The Quadratic Form Case

In general it is difficult to make statements about the gestalt of annihilating polynomials of quadratic forms. But it is possible to give full sets of generators for the annihilating ideal.

Proposition

There exist monic polynomials Q0 = 1, Q1, . . . , Qr ∈ Z[X] and k0, . . . , kr−1, kr = 0 ∈ N0 with (i) k0 > · · · > kr−1 > kr and (ii) deg(Q0) < deg(Q1) < · · · < deg(Qr) such that {2k0Qx, 2k1Q1Qx, . . . , 2kr−1Qr−1Qx, QrQx} forms a full set of generators for Annx.

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Examples

Examples

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 27 / 43

Examples

First Examples

Let ϕ be a quadratic form over K of dimension n. K Pythagorean. Then W (K) and W (K) are torsion free. = ⇒ We have Q[ϕ]([ϕ]) = 0 and Q{ϕ}({ϕ}) = 0 and therefore Ann[ϕ] = (Q[ϕ]) and Ann{ϕ} = (Q{ϕ}). K not formally real. Then an element of W (K) is torsion if and only if its dimension is 0. Furthermore any element of W (K) is torsion. = ⇒ Qϕ = X − n ∈ Z[X] and Q{ϕ} = 1.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 28 / 43

Examples

The Real Numbers

Let ϕ be a quadratic form over R with n = dim(ϕ). It is well known, that Hom( W (R), Z) = {dim, sign} and Hom(W (R), Z) = {sign}. Set s := sign(ϕ). Since R is Pythagorean, we obtain Ann[ϕ] =

• (X − n)

if s = n, ((X − s)(X − n))

• therwise,

and Ann{ϕ} = (X − s).

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 29 / 43

Examples

Local Fields

Let K be a local field with finite residue field. It is possible to completely classify the elements of W (K) with the help of the following three invariants.

1 The dimension:

dim : W (K) − → Z.

2 The discriminant:

d : W (K) − → K ∗/(K ∗)2, [ϕ] − → (−1)

n(n−1) 2

det(ϕ), where n is the dimension of ϕ.

3 The Clifford invariant (also called Witt invariant):

c : W (K) − → 2Br(K).

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 30 / 43

Examples

Local Fields

With the help of calculations involving these three invariants we can describe the annihilating ideal of any given quadratic form over K.

Proposition

Let ϕ be a quadratic form over K, n := dim(ϕ), ϕ ∼ = n × 1. Then a complete and minimal set of generators for Ann[ϕ] ⊂ Z[X] is given as follows: Ann[ϕ] =     

• 2(X − n), (X − n)2

if d(ϕ) is a sum of two squares in K,

• 4(X − n), (X − n + 2)(X − n)
• therwise.

If ϕ ∼ = n × 1, then Ann[ϕ] = (X − n).

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 31 / 43

Examples

Remark

Now the previous examples and the Hasse-Minkowski Theorem can be used to obtain a similar result for global fields. Furthermore it is of course possible to obtain analogous results for Ann{ϕ}, where ϕ is a quadratic form over a local or global field K. But these results demand for even more case distinctions and do not yield any additional knowledge. Therefore we leave them out here.

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Annihilating Polynomials of Excellent Forms

Annihilating Polynomials of Excellent Forms

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Annihilating Polynomials of Excellent Forms

Notation

Two quadratic forms ϕ and ψ over K are called similar if there exists an a ∈ K ∗ such that ϕ ∼ = aψ. A quadratic form ψ over K is called a subform of a form ϕ over K if there exists a form χ over K such that ϕ ∼ = ψ⊥χ.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 34 / 43

Annihilating Polynomials of Excellent Forms

Pfister Forms

Definition

A quadratic form ϕ over K is called a k-fold Pfister form, k ∈ N0, if there exist a1, . . . , ak ∈ K ∗ such that ϕ ∼ = 1, a1 ⊗ · · · ⊗ 1, ak. We write a1, . . . , ak :=

k

• i=1

1, ai.

Note

A Pfister form ϕ is isotropic if and only if it is hyperbolic. In other words: A Pfister form is either anisotropic or hyperbolic.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 35 / 43

Annihilating Polynomials of Excellent Forms

Annihilating Polynomials of Pfister Forms

We use Hurrelbrink’s method. Let ϕ ∼ = b1, a1 ⊗ · · · ⊗ 1, ak be similar to a k-fold Pfister form over K, k ∈ N0. Then f := b(1 + a1) · · · (1 + ak) ∈ Z[G] is a preimage of [ϕ] in Z[G], where G = K ∗/(K ∗)2. Let χ ∈ Hom(Z[G], Z). We have χ(f ) =        2k if χ(ai) = 1 for all i, and χ(b) = 1, −2k if χ(ai) = 1 for all i, and χ(b) = −1,

• therwise.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 36 / 43

Annihilating Polynomials of Excellent Forms

Annihilating Polynomials of Pfister Forms

As a consequence we obtain the following result.

Theorem

For k ∈ N0 the polynomial (X − 2k)X(X + 2k) = X(X 2 − 22k) ∈ Z[X] annihilates [ϕ] and {ϕ}, where ϕ is similar to a k-fold Pfister form

• ver any field K.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 37 / 43

Annihilating Polynomials of Excellent Forms

Pfister Neighbours

Definition

Let τ be a Pfister form over K. A quadratic form ϕ over K is called Pfister neighbour of τ if ϕ is similar to a subform of τ, and dim(ϕ) > 1

2 dim(τ).

Let ϕ be a Pfister neighbour of τ. Then there exists an a ∈ K ∗ and a form ψ over K such that aτ ∼ = ϕ⊥ψ. The form ψ is called the complement of ϕ.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 38 / 43

Annihilating Polynomials of Excellent Forms

Annihilating Polynomials of Pfister Neighbours

Theorem

Let ϕ be a Pfister neighbour over K with complement ψ. Let Q be an annihilating polynomial of [ψ] (resp. {ϕ}). Then Q · (X 2 − n2) ∈ Z[X] is an annihilating polynomial of [ϕ] (resp. {ϕ}).

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 39 / 43

Annihilating Polynomials of Excellent Forms

Excellent Quadratic Forms

Definition

Forms of dimension 0 and 1 are excellent. If ϕ is a form over K with dim(ϕ) > 1, then ϕ is excellent if ϕ is a Pfister neighbour whose complement is excellent. Let ϕ be an excellent form over K. Then there exists a sequence of forms ϕ = χ0, χ1, . . . , χr such that χi−1 is a Pfister neighbour with complement χi for i = 1, . . . , r, and dim(χr) ∈ {0, 1}.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 40 / 43

Annihilating Polynomials of Excellent Forms

Excellent Quadratic Forms

Let ϕ be excellent over K, and let χ0, . . . , χr be the sequence as defined above.

Definition

The form χk is called the k-th complement of ϕ.

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 41 / 43

Annihilating Polynomials of Excellent Forms

Annihilating Polynomials of Excellent Forms

Theorem

Let ϕ be an excellent form over K of dimension n, and let n = n0 > · · · > nr be the dimensions of the higher complements of ϕ. Then En :=

• X(X 2 − n2

r−1) · · · (X 2 − n2 1)(X 2 − n2)

for n even, (X 2 − 12)(X 2 − n2

r−1) · · · (X 2 − n2 1)(X 2 − n2)

for n odd, is an annihilating polynomial of [ϕ] and {ϕ}.

Remark

There exists a field K such that for any n ∈ N0 there exists an excellent form ϕ over K with Ann[ϕ] = Ann{ϕ} = (En).

Klaas-Tido R¨ uhl (EPFL) Annihilating Polynomials of Excellent Forms GTEM - Cetraro 2008 42 / 43

The End.

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