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Abstract We derive, via the Hardy–Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of nonsingular local solubility. Our polynomials satisfy the condition that . Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in many circumstances the expected asymptotic formula holds when and .
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A positive proportion of locally soluble quartic Thue equations are globally insoluble
Abstract For any fixed nonzero integer h , we show that a positive proportion of integral binary quartic forms F do locally everywhere represent h , but do not globally represent h . We order classes of integral binary quartic forms by the two generators of their ring of $${\rm GL}_{2}({\mathbb Z})$$ -invariants, classically denoted by I and J .
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- Award ID(s):
- 2001281
- PAR ID:
- 10334637
- Date Published:
- Journal Name:
- Mathematical Proceedings of the Cambridge Philosophical Society
- ISSN:
- 0305-0041
- Page Range / eLocation ID:
- 1 to 16
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0305004121000554
