Frobenius dilation of Hilbert function, Demailly's conjecture, and varieties of powers
Abstract
We prove a Frobenius-dilation inequality for Hilbert functions of ideals generated by powers of fixed homogeneous forms in characteristic zero. Via Emsalem--Iarrobino inverse system duality, it yields quantitative fat-point postulation estimates and a new proof of the recent theorem of Ha--Sivakumar resolving Demailly's conjecture for arbitrary finite point sets. A fixed-characteristic version controls tangent spaces to varieties of powers and, together with plane interpolation and secant-theoretic results, proves complete nondefectivity of Vk2,d for all k≥3 and d≥2; in particular, it settles the ternary Waring problem for generic k-rank conjectured by Lundqvist--Oneto--Reznick--Shapiro (suggested by Ottaviani). We also show that, for a prime power exponent q, any failure of Nicklasson's conjecture in three variables is confined to 2q−2 consecutive degrees.
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BibTeX
@misc{dinh2026frobenius,
title = {{Frobenius dilation of Hilbert function, Demailly's conjecture, and varieties of powers}},
author = {Trung Hoa Dinh and Tai Huy Ha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06767},
eprint = {2609.06767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06767},
}
RIS
TY - UNPB
TI - Frobenius dilation of Hilbert function, Demailly's conjecture, and varieties of powers
AU - Trung Hoa Dinh
AU - Tai Huy Ha
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06767
UR - https://arxiv.org/abs/2609.06767
AB - We prove a Frobenius-dilation inequality for Hilbert functions of ideals generated by powers of fixed homogeneous forms in characteristic zero. Via Emsalem--Iarrobino inverse system duality, it yields quantitative fat-point postulation estimates and a new proof of the recent theorem of Ha--Sivakumar resolving Demailly's conjecture for arbitrary finite point sets. A fixed-characteristic version controls tangent spaces to varieties of powers and, together with plane interpolation and secant-theoretic results, proves complete nondefectivity of V^k_2,d for all k≥3 and d≥2; in particular, it settles the ternary Waring problem for generic k-rank conjectured by Lundqvist--Oneto--Reznick--Shapiro (suggested by Ottaviani). We also show that, for a prime power exponent q, any failure of Nicklasson's conjecture in three variables is confined to 2q−2 consecutive degrees.
ER -