Anisotropic Quadratic Form Research Articles

Fields with bounded Brauer 2-torsion index

It is shown that, over a field of characteristic not $2$, the dimension of an anisotropic quadratic Pfister form of trivial total signature is at most twice the dimension of some central division algebra of exponent $2$. The proof is based on computations with quadratic trace forms of central simple algebras.

An ultimate proof of Hoffmann–Totaro’s conjecture

We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic \ne2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the author in 2002. In characteristic 2 , the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.

Some versions of the U-invariant of a field

Let F be a field, char F ≠ 2 . Assume that a 1 , … a n ∈ F ⁎ are such that a ‾ 1 , … a ‾ n ∈ F ⁎ / F ⁎ 2 are linearly independent over Z / 2 Z . As usual W ( F ) stands for the Witt ring of F . For an element φ ∈ W ( F ) denote by dim φ the dimension of the corresponding anisotropic quadratic form . Define u ˆ ( F ; a 1 , … , a n ) as the maximum of dim φ , where φ runs over the set of elements in W ( F ) , which become zero in W ( F ( a 1 , … , a n ) ) . This is a version of the classical notion of the u -invariant u ( F ) of the field F . It turns out that u ˆ ( F ; a 1 , … , a n ) ≤ α n ∑ i = 1 n u ˆ ( F ; a i ) for any n ≥ 2 , where the sequence α n is defined recurrently as α 2 = 1 , and α n = 5 2 ( n − 1 ) α n − 1 + 1 n . We compute u ˆ ( F ; a ) in certain cases, and show that u ˆ ( F ( b ) ; a ) ≤ 5 2 u ˆ ( F ; a ) , where b ∈ F ⁎ . However, in general there is no lower bound for u ˆ ( F ( b ) ; a ) via u ˆ ( F ; a ) , even though we prove that max ⁡ { u ˆ ( F ( b ) ; a ) , u ˆ ( F ( a b ) ; a ) } ≥ 1 3 u ˆ ( F ; a ) . Let u ˆ ( F ) be the maximum of u ˆ ( F ; a ) , where a runs over all elements of F ⁎ ∖ F ⁎ 2 . We show that u ˆ ( F ( b ) ) ≥ 1 4 u ˆ ( F ) if b is a sum of two squares. In particular, the last inequality holds if − 1 ∈ F ⁎ .

Similarity of quadratic and symmetric bilinear forms in characteristic 2

We say that a field extension L/F has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over F that become isometric (resp. similar) over L are already isometric (resp. similar) over F. The famous Artin–Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic 2. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic 2, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau’s norm principle for arbitrary quadratic or bilinear forms in characteristic 2.

On polynomial values of anisotropic quadratic forms

Let k be a field, chark≠2, let φ be an anisotropic quadratic form over k, dimφ≥2, V the underlying linear space of φ. As usual, denote by D(φ) the set of nonzero values of φ. Given a positive integer m, we say that φ is m-essential if there exists a nonzero polynomial p∈k[x1,…,xm] such that p∈D(φk(x1,…,xm)), but p∉D(ψk(x1,…,xm))) for any anisotropic form ψ over k with dimψ<dimφ. We say that φ is strongly m-essential if additionally p=φ(u) for some vector u∈V⊗kk[t1,…,tm]. Among other results we show that if chark=0, b∈k⁎, and the form π≃《−1,b》 is anisotropic, then π is 2-essential. If, additionally, −3∈k⁎, then π is strongly 2-essential. We prove also that any anisotropic 2-fold Pfister form over k becomes strongly 2-essential over a certain field extension of k. To prove these statements we consider the family of polynomialspa,b,ω(x,y)=−bx4y2+b2x2y4−b(a+2ω)x2y2−(ω+a)416a in two variables over k, which are generalizations of the Motzkin polynomial f(x,y)=x2y4+x4y2−3x2y2+1, since f(x,y)=p−1,−1,−1(x,y). We show that pa,b,ω∈D(《a,b》k(x,y)), but, on the other hand, p−1,b,−1∉D(φ)k(x,y) for each 3-dimensional anisotropic form φ over k, provided chark=0. The result is a generalization of a theorem of Cassels, Ellison and Pfister, which claims that the Motzkin polynomial f(x,y) is not a sum of three squares over R(x,y).Finally, at the end of the paper we give examples showing that the Cassels-Pfister theorem in two variables does not hold in general for n-dimensional anisotropic forms provided n≥4.

Quasilinear quadratic forms and function fields of quadrics

Let p and q be anisotropic quadratic forms of dimension $$\ge 2$$ over a field F. In a recent article, we formulated a conjecture describing a general constraint which the dimensions of p and q impose on the isotropy index of q after scalar extension to the function field of p. This can be viewed as a generalization of Hoffmann’s Separation Theorem which simultaneously incorporates and refines some well-known classical results on the Witt kernels of function fields of quadrics. Using algebro-geometric methods, it was shown that large parts of this conjecture hold in the case where the characteristic of F is not 2. In the present article, we prove similar (in fact, somewhat sharper) results in the case where F has characteristic 2 and q is a so-called quasilinear form. In contrast to the situation where $$\mathrm {char}(F) \ne 2$$, the methods used to treat this case are purely algebraic.

Quasilinear convexity and quasilinear stars in the ray space of a supertropical quadratic form

Relying on rays, we search for submodules of a module V over a supertropical semiring on which a given anisotropic quadratic form is quasilinear. Rays are classes of a certain equivalence relation on V, that carry a notion of convexity, which is consistent with quasilinearity. A criterion for quasilinearity is specified by a Cauchy-Schwartz ratio which paves the way to a convex geometry on , supported by a ‘supertropical trigonometry’. Employing a (partial) quasiordering on , this approach allows for producing convex quasilinear sets of rays, as well as paths, which contain a given quasilinear set in a systematic way. Minimal paths are endowed with a surprisingly rich combinatorial structure, delivered to the graph determined by pairs of quasilinear rays – apparently a fundamental object in the theory of supertropical quadratic forms.

Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics

Let p and q be anisotropic quadratic forms over a field F of characteristic $$\ne 2$$ , let s be the unique non-negative integer such that $$2^s < {\mathrm {dim}( p)} \le 2^{s+1}$$ , and let k denote the dimension of the anisotropic part of q after scalar extension to the function field F(p) of p. We conjecture that the dimension of q must lie within k of an integer multiple of $$2^{s+1}$$ . This can be viewed as a direct generalization of Hoffmann’s Separation theorem. Among other cases, we prove that the conjecture is true if $$k<2^{s-1}$$ . When $$k=0$$ , this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism $$W(F)\rightarrow W(F(p))$$ has dimension divisible by $$2^{s+1}$$ .

A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

Let$q$be an anisotropic quadratic form defined over a general field$F$. In this article, we formulate a new upper bound for the isotropy index of$q$after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where$\text{char}(F)\neq 2$, and (ii) the case where$\text{char}(F)=2$and$q$isquasilinear(i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

Norms and gauges on Clifford Algebra

ABSTRACTFrom a normed quadratic space (V,q), we construct a norm on the Clifford algebra C(V,q). We describe the associated graded form of this norm and give a condition for this norm to be a gauge. Then, we apply our results to prove that for a complete discrete valued field, an anisotropic quadratic form q with dimq = 0 mod 8 and nonsplit Clifford algebra cannot be at the same time a transfer of a K-hermitian form with K∕F an inertial quadratic field extension and a transfer of a T-hermitian form with T∕F a ramified quadratic field extension.

Generalised quadratic forms and the u-invariant

The u-invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u-invariants for hermitian and generalised quadratic forms over a division algebra with involution in characteristic 2 and investigate the relationships between them. We investigate these invariants in particular in the case of a quaternion algebra and further when this quaternion algebra is the unique quaternion division algebra over a field.

A reduction theorem for primitive binary permutation groups

A permutation group $(X,G)$ is said to be binary, or of relational complexity $2$, if for all $n$, the orbits of $G$ (acting diagonally) on $X^2$ determine the orbits of $G$ on $X^n$ in the following sense: for all $\bar{x},\bar{y} \in X^n$, $\bar{x}$ and $\bar{y}$ are $G$-conjugate if and only if every pair of entries from $\bar{x}$ is $G$-conjugate to the corresponding pair from $\bar{y}$. Cherlin has conjectured that the only finite primitive binary permutation groups are $S_n$, groups of prime order, and affine orthogonal groups $V\rtimes O(V)$ where $V$ is a vector space equipped with an anisotropic quadratic form; recently he succeeded in establishing the conjecture for those groups with an abelian socle. In this note, we show that what remains of the conjecture reduces, via the O'Nan-Scott Theorem, to groups with a nonabelian simple socle.

On the 3-Pfister Number of Quadratic Forms

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F×2-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant, and Hasse–Witt invariant can be written in the Witt ring as the sum of at most (d2)/8 3-fold Pfister forms.

Noise suppression in reconstruction of low‐Z target megavoltage cone‐beam CT images

To improve the image contrast-to-noise (CNR) ratio for low-Z target megavoltage cone-beam CT (MV CBCT) using a statistical projection noise suppression algorithm based on the penalized weighted least-squares (PWLS) criterion. Projection images of a contrast phantom, a CatPhan(®) 600 phantom and a head phantom were acquired by a Varian 2100EX LINAC with a low-Z (Al) target and low energy x-ray beam (2.5 MeV) at a low-dose level and at a high-dose level. The projections were then processed by minimizing the PWLS objective function. The weighted least square (WLS) term models the noise of measured projection and the penalty term enforces the smoothing constraints of the projection image. The variance of projection data was chosen as the weight for the PWLS objective function and it determined the contribution of each measurement. An anisotropic quadratic form penalty that incorporates the gradient information of projection image was used to preserve edges during noise reduction. Low-Z target MV CBCT images were reconstructed by the FDK algorithm after each projection was processed by the PWLS smoothing. Noise in low-Z target MV CBCT images were greatly suppressed after the PWLS projection smoothing, without noticeable sacrifice of the spatial resolution. Depending on the choice of smoothing parameter, the CNR of selected regions of interest in the PWLS processed low-dose low-Z target MV CBCT image can be higher than the corresponding high-dose image. The CNR of low-Z target MV CBCT images was substantially improved by using PWLS projection smoothing. The PWLS projection smoothing algorithm allows the reconstruction of high contrast low-Z target MV CBCT image with a total dose of as low as 2.3 cGy.

The Kneser-Tits conjecture for groups with Tits-index $$ {\text{E}}_{8,2}^{66} $$ over an arbitrary field

We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q_E is hyperbolic. (2) If G is the group of K-rational points of an absolutely simple algebraic group whose Tits index is E_{8,2}^{66}, then G is generated by its root groups, as predicted by the Kneser–Tits conjecture.

TH-E-220-03: Noise Suppression in Reconstruction Low-Z Target MV CBCT Images

Purpose: To improve the image contrast‐to‐noise (CNR) ratio for low‐Z target MV CBCT while preserving the spatial resolution using a statistical projection noise suppression algorithm based on penalized weighted least‐ squares (PWLS) criterion. Methods: Projection images of a contrast phantom was acquired by a Varian 2100EX LINAC with a low‐Z (Al) target and low energy x‐ray beam (2.5 MeV) at two doses levels, 2.3 cGy for low dose and 23.1cGy for high dose, measured at the center of phantom. The projections of low‐Z target MV CBCT were then processed by minimizing the penalized weighted least‐squares (PWLS) objective function. The WLS term models the noise of measured projection and the penalty term enforces the smoothing constraints of the projection image. The variance of projection data was chosen as the weight for the PWLS objective function and it determined the contribution of each measurement. An anisotropic quadratic form penalty that incorporates the gradient information of projection image was used to preserve edges during noise reduction. Low‐Z target MV CBCTimages were reconstructed by FDK after each projection was processed by the PWLS smoothing. Results: Noise in low‐Z target MV CBCTimages was greatly suppressed after the PWLS projection smoothing, without noticeable sacrifice of the spatial resolution. Contrast‐to‐noise ratio (CNR) of the selected regions of interest in the PWLS processed low‐Z target MV CBCTimage is even higher than the corresponding high‐dose image, and is 8 times higher than that of un‐processed low‐dose images. Conclusions: The CNR of low‐Z target MV CBCTimages was substantially improved while the spatial resolution was preserved by using PWLS projection smoothing. The PWLS projection smoothing algorithm allows low‐Z target MV CBCT radiation dose reduction by a factor of 10 without compromising the quality of reconstructed images.

The orthogonal $u$-invariant of a quaternion algebra

In quadratic form theory over fields, a much studied field invariant is the $u$-invariant, defined as the supremum of the dimensions of anisotropic quadratic forms over the field. We investigate the corresponding notions of $u$-invariant for hermitian and for skew-hermitian forms over a division algebra with involution, with a special focus on skew-hermitian forms over a quaternion algebra with canonical involution. Under certain conditions on the center of the quaternion algebra, we obtain sharp bounds for this invariant.

Birational composition of quadratic forms over a local field

Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n, respectively, over K, char K ≠ 2. The problem of birational composition of f(X) and g(Y) is considered: When is the product f(X) · g(Y) birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The solution of the birational composition problem for anisotropic quadratic forms over K in the case of m = n = 2 is given. The main result of the paper is the complete solution of the birational composition problem for forms f(X) and g(Y) over a local field P, char P ≠ 2.

ON HERMITIAN PFISTER FORMS

Let K be a field of characteristic different from 2. It is known that a quadratic Pfister form over K is hyperbolic once it is isotropic. It is also known that the dimension of an anisotropic quadratic form over K belonging to a given power of the fundamental ideal of the Witt ring of K is lower bounded. In this paper, weak analogues of these two statements are proved for hermitian forms over a multiquaternion algebra with involution. Consequences for Pfister involutions are also drawn. An invariant uα of K with respect to a nonzero pure quaternion of a quaternion division algebra over K is defined. Upper bounds for this invariant are provided. In particular an analogue is obtained of a result of Elman and Lam concerning the u-invariant of a field of level at most 2.

WE-C-351-05: Dose Reduction in Kilovotage Cone-Beam Computed Tomography for Radiation Therapy

Purpose: To develop an effective method of improving low‐dose kV CBCTimage quality using statistics‐based sinogram smoothing so that the patient's imagingdose can be greatly reduced through the use of low‐mAs protocol. Method and Materials: Based on the sinogramnoise properties, a penalized weighted least‐squares (PWLS) objective function was constructed, and the ideal sinogram was then estimated by minimizing the PWLS objection function. The variance of sinogram data was chosen as the weight the PWLS objective function and it determined the contribution of each measurement. To preserve edge information during noise reduction, we proposed an anisotropic quadratic form penalty. The quadratic form penalty encourages equivalence between neighbors and the anisotropic penalty provides the mechanism to control the influence of different neighbors according to its corresponding gradient. The proposed anisotropic penalty tends to discourage equivalence between neighbors if the gradient is large, thus edge information will be preserved in the smoothed sinogram. Two experimental phantom studies were performed to demonstrate the effectiveness of the presented algorithm. CBCT projection images were acquired by the ExactArms (kV source/detector arms) of a Varian Trilogy™ treatment system. Results:Noise in the reconstructedCBCTimage acquired with the low‐mAs protocol was greatly suppressed after the proposed sinogramimage processing, without noticeable sacrifice of the spatial resolution. For both phantoms, image quality of the low‐dose (100 mAs/projection) CBCTreconstructed from the sinogram processed by the PWLS criterion is comparable to its corresponding normal‐dose (800 mAs/projection) images in terms of contrast‐to‐noise ratio and fine structures preservation. Conclusion: The presented PWLS sinogram smoothing algorithm reduces CBCT radiation dose by a factor of 1/8 without compromising the quality of reconstructed images.