Abstract.: If φ is an excellent form, then it is possible to use the dimensions of the higher complements of φ to obtain an annihilating polynomial of φ of low degree. The main result of this paper is the construction of such a polynomial with the help of methods from the theory of generic splitting of quadratic form

Rühl, Klaas-Tido

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Arch. Math. 90 (2008), 217–222c© 2008 Birkha¨user Verlag Basel/Switzerland0003/889X/030217-6, published online 2008-02-14DOI 10.1007/s00013-007-2410-1 Archiv der MathematikAnnihilating polynomials of excellent quadratic formsKlaas-Tido Ru¨hlAbstract. If ϕ is an excellent form, then it is possible to use the dimensions ofthe higher complements of ϕ to obtain an annihilating polynomial of ϕ of lowdegree. The main result of this paper is the construction of such a polynomialwith the help of methods from the theory of generic splitting of quadraticforms.Mathematics Subject Classiﬁcation (2000). Primary 11E81; Secondary 16R99.Keywords. Quadratic form, annihilating polynomial.1. Introduction. In [5] Lewis showed that for each n ∈ N0 the polynomialZ[X]  Pn :={X(X2 − 2) · · · (X2 − (n − 2)2)(X2 − n2) for n even(X2 − 1)(X2 − 32) · · · (X2 − (n − 2)2)(X2 − n2) for n oddannihilates every n-dimensional quadratic form over an arbitrary ﬁeld K. Sincethen a number of classes of quadratic forms have been identiﬁed for which anni-hilating polynomials of signiﬁcantly lower degree can be constructed, e.g. positiveforms and trace forms. An extensive survey of the currently known results aboutannihilating polynomials of quadratic forms can be found in [6].In this paper we will consider the class of excellent forms. Let ϕ be an excellentform of dimension n over K, let h be its height, and for j = 0, . . . , h let nj be thedimension of the j-th complement of ϕ. We will show thatZ[X]  En :={X(X2 − n2h−1) · · · (X2 − n21)(X2 − n2) for n even(X2 − 1)(X2 − n2h−1) · · · (X2 − n21)(X2 − n2) for n oddThe author gratefully acknowledges support from the Swiss National Science Foundation, grantno. 200020-109174/1 (project leader: E. Bayer-Fluckiger).218 K.-T. Ru¨hl Arch. Math.annihilates ϕ. If ϕ is isotropic, then we can prove this just by using the fact thatϕ is a Pﬁster neighbour of a hyperbolic Pﬁster form. If ϕ is anisotropic we will usemethods from the theory of generic splitting of quadratic forms.2. Notation and Basic Facts. The reader is expected to be familiar with thealgebraic theory of quadratic forms and the theory of generic splitting of qua-dratic forms. An introduction to quadratic forms can be found in [7], [8] or [4].The theory of generic splitting of quadratic forms was introduced in [2].We deﬁne the set of natural numbers N not to contain 0. If we need 0 as anatural number we use the notation N0. In all of this paper K will be a ﬁeld withchar(K) = 2, and we only consider non-degenerate quadratic forms over K. Oftenwe will simply use the notion “form” when we mean a quadratic form.Let ϕ be a quadratic form over K. We denote by dim(ϕ) the dimension of thequadratic space over K corresponding to ϕ. If ψ is another quadratic form overK, then we write ϕ⊥ψ and ϕ ⊗ ψ for the orthogonal sum and the tensor productof ϕ and ψ. In case there exists a form χ over K such that ϕ ∼= ψ⊥χ we callψ a subform of ϕ. For n ∈ N0 the n-fold sum of ϕ will be denoted by n × ϕ. Ifϕ = 〈a1, . . . , an〉 with a1, . . . , an ∈ K∗, then aϕ := 〈aa1, . . . , aan〉 for a ∈ K∗. Inparticular, if a = −1, we write −ϕ := 〈−a1, . . . ,−an〉. For a1, . . . , ak ∈ K∗ we usethe notation 〈〈a1, . . . , ak〉〉 for the k-fold Pﬁster form⊗ki=1〈1, ai〉. If there existssome a ∈ K∗ such that ϕ ∼= aψ we say that ϕ and ψ are similar. We denote byŴ (K), respectively W (K), the Witt-Grothendieck ring, respectively Witt ring, ofK. We write [ϕ] for the isometry class of ϕ in Ŵ (K), and if ϕ is isometric to ψwe use the notation ϕ ∼= ψ. In the case that ϕ ∼= n×〈1〉 we simply write n insteadof [n × 〈1〉] for the isometry class of ϕ in Ŵ (K). The sum respectively product of[ϕ] and [ψ] in Ŵ (K) will be denoted by [ϕ] + [ψ] respectively [ϕ] [ψ] (or [ϕ] · [ψ]).For an arbitrary element [ϕ]− [ψ] ∈ Ŵ (K) we can extend the notion of dimensionby setting dim([ϕ] − [ψ]) := dim(ϕ) − dim(ψ). If the forms ϕ and ψ represent thesame element in W (K) they are called equivalent, and we write ϕ ∼ ψ. For anelement a ∈ K∗ its class in the square class group K∗/(K∗)2 will be denoted bya := a(K∗)2.If L is a ﬁeld extension over K, then we write ϕL for the form ϕ consideredas a quadratic form over L. Most important in this paper is the case that L isthe function ﬁeld K(ϕ) of ϕ. To be exact, if dim(ϕ) ≥ 2 and ϕ ∼= 〈1,−1〉 wedenote by K(ϕ) the function ﬁeld of the quadric deﬁned by ϕ, otherwise we setK(ϕ) := K. Now consider a generic splitting tower K0 := K ⊂ K1 ⊂ · · · ⊂ Khof ϕ (see [3, Section 5]), then h is called the height of ϕ. For j = 0, . . . , h weset ϕj := (ϕKj )an, where (ϕKj )an denotes the anisotropic kernel of ϕKj , andcall ϕj the j-th anisotropic kernel of ϕ. The form ϕ is a Pﬁster neighbour ifthere exists a Pﬁster form τ over K such that ϕ is similar to a subform of τ anddim(ϕ) > 12 dim(τ). If χ is a form over K and a ∈ K∗ such that aτ ∼= ϕ⊥χ, then χis called the complement of ϕ. We say that ϕ is excellent if there exists a sequenceVol. 90 (2008) Annihilating polynomials of excellent quadratic forms 219ψ0 := ϕ,ψ1, . . . , ψh of forms over K such that dim(ψh) ≤ 1 and ψi−1 is a Pﬁsterneighbour with complement ψi for i = 1, . . . , h. The ψi are the higher complementsof ϕ. In this setting, if ϕ is anisotropic, it is well-known that (ψj)Kj ∼= (−1)jϕjfor j = 0, . . . , h (see [3, Proposition 7.9]).Deﬁnition 2.1. Let R be a unitary, commutative ring, and let ι : Z → S be thecanonical homomorphism deﬁned by ι(1) = 1S . If P = anXn+. . . a1X+a0 ∈ Z[X]is a polynomial and x ∈ S, then P is called an annihilating polynomial of x ifP (x) := ι(an)xn + · · · + ι(a1)x + ι(a0) = 0.Usually we will omit the ι and just write as := ι(a)s for a ∈ Z and s ∈ S.Let G := K∗/(K∗)2 be the square class group of K. We use Hurrelbrink’sapproach in [1] to construct annihilating polynomials for elements of the groupring Z[G]:Consider the dual groupĜ :={χ : G → S1 | χ is a group homomorphism}of G, where S1 ⊂ C is the unit circle. Then each character χ ∈ Ĝ can be extendedto a Z-algebra homomorphismχ : Z[G] −→ C,∑g∈Gzgg −→∑g∈Gzgχ(g).For f ∈ Z[G] the set Sf :={χ(f)∣∣ χ ∈ Ĝ} is ﬁnite by [1, Observation 1.2], and itis easy to see that it is a subset of Z. This allows us to deﬁne the polynomialPf :=∏χ(f)∈Sf(X − χ(f)) ∈ Z[X],which by [1, Theorem 1.3] is an annihilating polynomial of f .Now consider the canonical surjection π1 : Z[G] → Ŵ (K) deﬁned bya1 + · · · + an −→ 〈a1, . . . , an〉.Deﬁnition 2.2. An element f =∑g∈G zgg ∈ Z[G] is called a preform if zg ≥ 0for all g ∈ G. The integer n := ∑g∈G zg is the dimension of f , and we writedim(f) := n. If f = a1 + · · · + an, and if ϕ is a quadratic form over K, then f iscalled a preform of ϕ if we have ϕ ∼= 〈a1, . . . , an〉 or equivalently π1(f) = [ϕ].Deﬁnition 2.3. Let ϕ be a quadratic form over K. A polynomial P ∈ Z[X] is calledan annihilating polynomial of ϕ if it is an annihilating polynomial of [ϕ] ∈ Ŵ (K).Corollary 2.4. If f ∈ Z[G] is a preform of a quadratic form ϕ over K, then Pf isan annihilating polynomial of ϕ.220 K.-T. Ru¨hl Arch. Math.Proof. We havePf([ϕ])= Pf(π1(f))= π1(Pf (f))= π1(0) = 0.Remark 2.5. If f ∈ Z[G] is a preform of an arbitrary form ϕ over K, dim(ϕ) = n,then it is easy to see that Sf ⊂ {−n,−n + 2, . . . , n − 2, n} (see [1, Observation2.2]). We thus obtain the result ﬁrst presented by Lewis in [5], that the polynomialPn := (X + n)(X + n − 2) · · · (X − n + 2)(X − n) ∈ Z[X]annihilates all n-dimensional quadratic forms over K.3. Main Result.Lemma 3.1. If ϕ and ψ are quadratic forms over K with ϕ ∼ ψ, and if P ∈ Z[X]is an annihilating polynomial of ϕ, then P ([ψ]) ∼ 0.Proof. Let π2 : Ŵ (K) → W (K) be the natural projection. Then0 = π2(P ([ϕ])) = P (π2([ϕ])) = P (π2([ψ])) = π2(P ([ψ]))or equivalently P ([ψ]) ∼ 0. Proposition 3.2. Let ϕ be a Pﬁster neighbour of dimension n > 0 with complement−ψ over K, and let a ∈ K∗ and τ be a Pﬁster form such that ϕ⊥ − ψ ∼= aτ . IfQ ∈ Z[X] is an annihilating polynomial of ψ, then P := Q · (X2 − n2) is anannihilating polynomial of ϕ. In the case that a ∈ (K∗)2 (in particular if ϕ isisotropic) the polynomial P˜ := Q · (X − n) suﬃces.Proof. If ϕ is isotropic, then τ is isotropic as well and therefore hyperbolic. Henceϕ ∼ ψ. By the previous lemma Q([ϕ]) ∼ 0, and thereforeQ([ϕ]) · [ϕ] ∼ 0 ∼ Q([ϕ]) · n.The claim follows since dim(Q([ϕ]) · [ϕ]) = dim(Q([ϕ]) · n).Now let ϕ be anisotropic. Since ϕK(ϕ) ∼ ψK(ϕ), we obtain Q([ϕK(ϕ)]) ∼ 0from the previous lemma. A result by Knebusch ([2, Example 4.1]) states thatQ([ϕK(τ)]) ∼ 0. Hence by [2, Lemma 4.4] there exists an element x ∈ Ŵ (K) suchthat Q([ϕ]) = x · [τ ]. Now aϕ ⊂ τ . Since bτ ∼= τ for all b ∈ K∗ represented by τ , itfollows that aϕ ⊗ τ ∼= n × τ . If a ∈ (K∗)2, thenQ([ϕ]) · ([ϕ] − n) = x · [τ ] · ([ϕ] − n) = x · ([ϕ ⊗ τ ] − [n × τ ]) = x · 0 = 0.Otherwise, if a is not a square, we still obtainQ([ϕ]) ·([ϕ]2 − n2)= x · [τ ] ·([ϕ]2 − n2)= x ·([τ ] · [aϕ]2 − [τ ] · n2)= x · ([aϕ ⊗ aϕ ⊗ τ ] − [n2 × τ]) = x · 0 = 0.Vol. 90 (2008) Annihilating polynomials of excellent quadratic forms 221Theorem 3.3. Let ϕ be an excellent form of height h ∈ N0 over K, let ϕ =ψ0, ψ1, . . . , ψh be the sequence of its higher complements, and for j = 0, . . . , hset nj := dim(ψj). ThenZ[X]  En :={X(X2 − n2h−1) · · · (X2 − n21)(X2 − n2) for n even(X2 − 1)(X2 − n2h−1) · · · (X2 − n21)(X2 − n2) for n oddannihilates ϕ.Proof. We proceed by induction on h. If h = 0 and dim(ϕ) = 0, then indeedE0 = X annihilates ϕ = 0. In the case dim(ϕ) = 1 we have ϕ = 〈a〉 for somea ∈ K∗, and [a]2 = 1. This shows, that E1 = X2 − 1 annihilates ϕ.Now let h > 0. By induction En1 annihilates the excellent form ψ1. Since ψ1 isthe complement of ϕ it follows from the previous lemma that En1 ·(X2 −n2) = Enannihilates ϕ. Remark 3.4. It is well-known that the dimensions of the higher complements andthe height of an excellent form ϕ only depend on the dimension of ϕ and can beexplicitly calculated with the help of certain recursive functions (see [3, Corollary7.11]).Remark 3.5. Let ϕ be similar to a k-fold Pﬁster form, k ∈ N0. A consequence ofproposition 3.2 is the well-known result that ϕ can be annihilated by X(X2 −22k)= X(X − 2k)(X + 2k) ∈ Z[X]. If ϕ is a k-fold Pﬁster form it suﬃces to take thepolynomial X(X − 2k).There exists an easy proof of this result that only uses corollary 2.4: Let ϕ besimilar to a k-fold Pﬁster form. Then there exist g0, g1, . . . , gk ∈ G := K∗/(K∗)2such that f = g0 · (1 + g1) · · · (1 + gk) ∈ Z[G] is a preform of ϕ. Since for everyχ ∈ Ĝ the map χ is a homomorphism of rings, and since χ(1 + gi) ∈ {0, 2} for alli = 1, . . . , k, it follows that χ(f) ∈ {−2k, 0, 2k} and therefore Sf ⊂ {−2k, 0, 2k}. Ifg0 = 1 we even get Sf ⊂ {0, 2k}.References[1] J. Hurrelbrink, Annihilating polynomials for group rings and Witt rings. Canad.Math. Bull. 32(4), 412–416 (1989).[2] M. Knebusch, Generic splitting of quadratic forms I. Proc. London Math. Soc. 33,65–93 (1976).[3] M. Knebusch, Generic splitting of quadratic forms II. Proc. London Math. Soc. 34,1–31 (1977).[4] T.-Y. Lam, Introduction to Quadratic Forms over Fields. American MathematicalSociety, 2005.[5] D. W. Lewis, Witt rings as integral rings. Invent. Math. 90, 631–633 (1987).222 K.-T. Ru¨hl Arch. Math.[6] D. W. Lewis, Annihilating polynomials for quadratic forms. Int. J. Math. Math. Sci.27(7), 449–455 (2001).[7] A. Pfister, Quadratic Forms with Applications to Algebraic Geometry and Topol-ogy, London Math. Soc. Lecture Note Series 217. Cambridge University Press, 1995.[8] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der math. Wis-senschaften 270. Springer-Verlag Berlin Heidelberg New York Tokyo, 1985.Klaas-Tido Ru¨hl, EPFL SB IMB CSAG, MA C3 575 (Batiment MA), Station 8,CH-1015 Lausanne, Switzerlande-mail: klaas-tido.ruhl@epfl.chReceived: 23 April 2007

Annihilating polynomials of excellent quadratic forms

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Annihilating polynomials of excellent quadratic forms

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