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Let $F$ be a field. Following the resolution of Milnor's conjecture relating the graded Witt ring of $F$ to its mod-2 Milnor $K$-theory, a major problem in the theory of symmetric bilinear forms is to understand, for any positive integer $n$, the low-dimensional part of $I^n(F)$, the $n$th power of the fundamental ideal in the Witt ring of $F$. In a 2004 paper, Karpenko used methods from the theory of algebraic cycles to show that if $\mathfrak{b}$ is a non-zero anisotropic symmetric bilinear form of dimension $< 2^{n+1}$ representing an element of $I^n(F)$, then $\mathfrak{b}$ has dimension $2^{n+1} - 2^i$ for some $1 \leq i \leq n$. When $i = n$, a classical result of Arason and Pfister says that $\mathfrak{b}$ is similar to an $n$-fold Pfister form. At the next level, it has been conjectured that if $n \geq 2$ and $i= n-1$, then $\mathfrak{b}$ is isometric to the tensor product of an $(n-2)$-fold Pfister form and a $6$-dimensional form of trivial discriminant. This has only been shown to be true, however, when $n = 2$, or when $n = 3$ and $\mathrm{char}(F) \neq 2$ (another result of Pfister). In the present article, we prove the conjecture for all values of $n$ in the case where $\mathrm{char}(F) =2$. In addition, we give a short and elementary proof of Karpenko's theorem in the characteristic-2 case, rendering it free from the use of subtle algebraic-geometric tools. Finally, we consider the question of whether additional dimension gaps can appear among the anisotropic forms of dimension $\geq 2^{n+1}$ representing an element of $I^n(F)$. When $\mathrm{char}(F) \neq 2$, a result of Vishik asserts that there are no such gaps, but the situation seems to be less clear when $\mathrm{char}(F) = 2$.

Let $p$ and $q$ be anisotropic quasilinear quadratic forms over a field $F$ of characteristic $2$, and let $i$ be the isotropy index of $q$ after scalar extension to the function field of the affine quadric with equation $p=0$. In this article, we establish a strong constraint on $i$ in terms of the dimension of $q$ and two stable birational invariants of $p$, one of which is the well-known "Izhboldin dimension", and the other of which is a new invariant that we denote $\Delta(p)$. Examining the contribution from the Izhboldin dimension, we obtain a result that unifies and extends the quasilinear analogues of two fundamental results on the isotropy of non-singular quadratic forms over function fields of quadrics in arbitrary characteristic due to Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the quasilinear case of a general conjecture previously formulated by the author, suggesting that a substantial refinement of this conjecture should hold.

In this paper we describe QUBIC, an experiment that will observe the polarized microwave sky with a novel approach, which combines the sensitivity of state-of-the art bolometric detectors with the systematic effects control typical of interferometers. QUBIC unique features are the so-called "self-calibration", a technique that allows us to clean the measured data from instrumental effects, and its spectral imaging power, i.e. the ability to separate the signal in various sub-bands within each frequency band. QUBIC will observe the sky in two main frequency bands: 150 GHz and 220 GHz. A technological demonstrator is currently under testing and will be deployed in Argentina during 2019, while the final instrument is expected to be installed during 2020., Comment: Proceedings of the 2018 ICNFP conference, Crete. Published by Universe arXiv admin note: text overlap with arXiv:1801.03730

Let $p$ and $q$ be anisotropic quadratic forms of dimension $\geq 2$ over a field $F$. In a recent article, we formulated a conjecture describing the general constraints 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, slightly stronger) results in the case where $F$ has characteristic $2$ and $q$ is a so-called quasilinear form. In contrast to the situation where $\mathrm{char}(F) \neq 2$, the methods used to treat this case are purely algebraic.

Let $p$ and $q$ be anisotropic quadratic forms over a field $F$ of characteristic $\neq 2$, let $s$ be the unique non-negative integer such that $2^s < \mathrm{dim}(p) \leq 2^{s+1}$, 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 $\mathrm{dim}(q)$ must lie within $k$ of a 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 $2^{s+1}$., Comment: Re-written; main result improved

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 a celebrated bound established in earlier work of Karpenko-Merkurjev and Totaro; on the other, 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 important cases: (i) the case where $\mathrm{char}(F) \neq 2$, and (ii) the case where $\mathrm{char}(F) = 2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic-geometric, and the second being purely algebraic.

Let $R$ be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over $R$ remain anisotropic after base change to any odd-degree finite \'{e}tale extension of $R$. This generalization of the classical Artin-Springer theorem (concerning the situation where $R$ is a field) was previously established in the case where all residue fields of $R$ are infinite by I. Panin and U. Rehmann. The more general result presented here permits to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic $\neq 2$ to the case where the ring has at least one residue field which is finite.

The Earth's magnetosphere is formed as a consequence of interaction between the planet's magnetic field and the solar wind, a continuous plasma stream from the Sun. A number of different solar wind phenomena have been studied over the past forty years with the intention of understanding and forecasting solar behavior. One of these phenomena in particular, Earth-bound interplanetary coronal mass ejections (CMEs), can significantly disturb the Earth's magnetosphere for a short time and cause geomagnetic storms. This publication presents a mission concept consisting of six spacecraft that are equally spaced in a heliocentric orbit at 0.72 AU. These spacecraft will monitor the plasma properties, the magnetic field's orientation and magnitude, and the 3D-propagation trajectory of CMEs heading for Earth. The primary objective of this mission is to increase space weather (SW) forecasting time by means of a near real-time information service, that is based upon in-situ and remote measurements of the aforementioned CME properties. The mission's secondary objective is to provide vital data to update scientific models. In-situ measurements are performed using a Solar Wind Analyzer instrumentation package and flux gate magnetometers, while coronagraphs execute remote measurements. Communication with the six identical spacecraft is realized via a deep space network consisting of six ground stations. They provide an information service that is in uninterrupted contact with the spacecraft, allowing for continuous SW monitoring. The data will be handled by a dedicated processing center before being forwarded to the SSA Space Weather Coordination Center who will manage the SW forecasting. The data processing center will additionally archive the data for the scientific community. The proposed concept mission allows for major advances in SW forecasting time and the scientific modelling of SW., Comment: 29 pages, 9 figures, accepted by the Journal of Space Weather and Space Climate

We introduce Milnor-Witt $K$-groups of local rings and show that the $n$th Milnor-Witt $K$-group of a local ring $R$ which contains an infinite field of characteristic not $2$ is the pull-back of the $n$th power of the fundamental ideal in the Witt ring of $R$ and the $n$th Milnor $K$-group of $R$ over the $n$th Milnor $K$-group of $R$ modulo $2$. This generalizes the work of Morel-Hopkins on Milnor-Witt $K$-groups of fields.

One of the most significant discrete invariants of a quadratic form $\phi$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behaviour of $\phi$ under scalar extension to arbitrary overfields of $k$. A similarly important, but more accessible variant of this notion is that of the Knebusch splitting pattern of $\phi$, which captures the isotropy behaviour of $\phi$ as one passes over a certain prescribed tower of $k$-overfields. In this paper, we determine all possible values of this latter invariant in the case where $\phi$ is totally singular. This includes an extension of Karpenko's theorem (formerly Hoffmann's conjecture) on the possible values of the first Witt index to the totally singular case. Contrary to the existing approaches to this problem (in the nonsingular case), our results are achieved by means of a new structural result on the higher anisotropic kernels of totally singular quadratic forms. Moreover, the methods used here readily generalise to give analogous results for arbitrary Fermat-type forms of degree $p$ over fields of characteristic $p>0$. Related problems concerning the Knebusch splitting of symmetric bilinear forms over fields of characteristic 2 and the full splitting of totally singular quadratic forms are also considered.

We study the splitting behaviour of quasilinear $p$-forms in the spirit of the theory of nondegenerate quadratic forms over fields of characteristic different from 2 using an analogue of M. Knebusch's generic splitting tower. Several new applications to the theory of quasilinear quadratic forms are given. Among them, we can mention an algebraic analogue of A. Vishik's theorem on "outer excellent connections" in the motives of quadrics, partial results towards a quasilinear analogue of N. Karpenko's theorem on the possible values of the invariant $i_1$, and a proof of a conjecture of D. Hoffmann on quadratic forms with maximal splitting in the quasilinear case., Comment: 29 pages

Let $X$ be an integral affine or projective hypersurface over a field $F$ of characteristic $p>0$, and let $F(X)$ denote its function field. In a recent article, Dolphin and Hoffmann obtained an explicit description of the kernel of the natural restriction homomorphism between the rings of absolute K\"{a}hler differentials of $F$ and $F(X)$, respectively. In this note, we examine the possibility of deriving an analogous result for mod-$p$ Milnor $K$-theory using the Bloch-Gabber-Kato theorem.

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known for the class of quasilinear quadrics. We provide new proofs here, because the original proofs do not immediately generalise for quasilinear hypersurfaces of higher degree.