Pairs of cubic forms in many variables

AbstractWe consider a system of R cubic forms in n variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided {n\geq 25R}, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular, we obtain the Hasse principle for systems of cubic forms in {25R} variables, previous work having required that {n\gg R^{2}}. One conjectures that {n\geq 6R+1} should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.

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Hydride-based antiperovskites with soft anionic sublattices as fast alkali ionic conductors

Nature Communications ◽

10.1038/s41467-020-20370-2 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Shenghan Gao ◽

Thibault Broux ◽

Susumu Fujii ◽

Cédric Tassel ◽

Kentaro Yamamoto ◽

...

Keyword(s):

Experimental Studies ◽

Ionic Conductivities ◽

Cubic Phase ◽

Ionic Conductors ◽

Large Size ◽

Migration Barriers ◽

Soft Phonon Mode ◽

Aliovalent Substitution ◽

Cubic Forms ◽

The Ideal

AbstractMost solid-state materials are composed of p-block anions, only in recent years the introduction of hydride anions (1s2) in oxides (e.g., SrVO2H, BaTi(O,H)3) has allowed the discovery of various interesting properties. Here we exploit the large polarizability of hydride anions (H–) together with chalcogenide (Ch2–) anions to construct a family of antiperovskites with soft anionic sublattices. The M3HCh antiperovskites (M = Li, Na) adopt the ideal cubic structure except orthorhombic Na3HS, despite the large variation in sizes of M and Ch. This unconventional robustness of cubic phase mainly originates from the large size-flexibility of the H– anion. Theoretical and experimental studies reveal low migration barriers for Li+/Na+ transport and high ionic conductivity, possibly promoted by a soft phonon mode associated with the rotational motion of HM6 octahedra in their cubic forms. Aliovalent substitution to create vacancies has further enhanced ionic conductivities of this series of antiperovskites, resulting in Na2.9H(Se0.9I0.1) achieving a high conductivity of ~1 × 10–4 S/cm (100 °C).

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$GL_{2}(O_K)$-invariant lattices in the space of binary cubic forms with coefficients in the number field $K$

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-2014-11978-2 ◽

2014 ◽

Vol 142 (7) ◽

pp. 2313-2325

Author(s):

Charles A. Osborne

Keyword(s):

Number Field ◽

Cubic Forms

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Simultaneous Additive Equations: Repeated and Differing Degrees

Canadian Journal of Mathematics ◽

10.4153/cjm-2016-006-4 ◽

2017 ◽

Vol 69 (02) ◽

pp. 258-283 ◽

Cited By ~ 1

Author(s):

Julia Brandes ◽

Scott T. Parsell

Keyword(s):

Hybrid Systems ◽

Existence Of Solutions ◽

Square Root ◽

Multiple Forms ◽

Quadratic Equations ◽

Asymptotic Formulae ◽

Cubic Forms

Abstract We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formulæ, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning–Heath–Brown, which give weak results when specialized to the diagonal situation, this is the first result on such “hybrid” systems. We also obtain specialized results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.

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