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Matthew Hase-Liu

Publications and source records attributed to Matthew Hase-Liu.

10 recordsLinked to original sources
arXiv2026

The asymptotic in Waring's problem over function fields beyond twice the degree

We prove the expected asymptotic in Waring's problem over $\mathbb F_q[T]$, with a power-saving error, whenever $n>2d$, the characteristic is greater than $(d-1)^2$, and $q$ satisfies an explicit lower bound. This range is sharp in general for the expected asymptotic uniformly in the target polynomial. Our main new input is an aggregate minor arc estimate: we count functionals according to the codimension of their associated singular loci and use intersection theory to bound the degrees of the resulting parameter spaces. In particular, if $n\ge (2+\varepsilon)d$, the required lower bound on $q$ is polynomial in $d$ of degree $2+4/\varepsilon$.

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arXiv2026

Betti bounds for spaces of curves on varieties and Manin's conjecture for quartic del Pezzo surfaces

We prove uniform exponential bounds for the compactly supported Betti numbers of spaces of morphisms from curves of fixed genus to projective varieties. For targets in a fixed projective space cut out by a prescribed number of equations of fixed degrees, the bound is exponential in the degree of the morphism and is independent of the ground field, the source curve, and the target. The proof constructs bounded-degree affine presentations involving only linearly many variables and equations, and then applies Katz's estimate. As an application, we establish a higher genus function field version of Manin's conjecture for split quartic del Pezzo surfaces, generalizing a recent theorem of Das--Lehmann--Tanimoto--Tosteson. Over sufficiently large finite fields, and after restricting curve classes to a slightly shrunken nef cone, we obtain the predicted asymptotic with the expected leading constant. As in Das--Lehmann--Tanimoto--Tosteson's argument, we combine the uniform Betti bound with a higher genus homological sieve, a bar complex calculation, and a virtual height zeta function.

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arXiv2026

Birch's theorem over function fields with quadratically many variables

For smooth hypersurfaces over rational function fields of characteristic greater than the degree and with sufficiently large constant field, we improve the number of variables required in Birch's theorem from an exponential function of the degree to a quadratic one. This agrees, up to constants, with the sharp quadratic threshold for the unconditional existence of rational points on smooth hypersurfaces. A circle method argument reduces the required cancellation to lower bounds for the codimensions of certain singular loci associated with complete exponential sums over finite fields. Our main innovation is a new method for proving these bounds: we introduce the notion of multiplication rank for the linear functionals indexing these exponential sums and combine the resulting rank stratification with a weighted degeneration of the Jacobian equations to obtain a codimension estimate that grows linearly with multiplication rank.

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arXiv2026

A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality

Geometric Manin's conjecture predicts that components of the moduli space of curves on a Fano variety parametrizing non-free curves are pathological and arise from "accumulating" morphisms that increase the Fujita invariant. By passing to positive characteristic and employing a higher genus generalization of the circle method, we prove a converse to this conjecture for general hypersurfaces $X$ in $\mathbb{P}^{n}$ of degree $d\le n/4+3/2$, namely that there are no such accumulating maps to $X$. One consequence of this is a version of Poincaré duality for these moduli spaces in a range.

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arXiv2025

A higher genus circle method and an application to geometric Manin's conjecture

Browning and Vishe used the Hardy-Littlewood circle method to show the moduli space of rational curves on smooth hypersurfaces of low degree is irreducible and of the expected dimension. We reinterpret the circle method geometrically and prove a generalization for higher genus smooth projective curves. In particular, we explain how the geometry of numbers can be understood via the Beauville-Laszlo theorem in terms of vector bundles on curves and their slopes, allowing us to prove a higher genus variant of Davenport's shrinking lemma. As a corollary, we apply this result to show the Fujita invariant of any proper subvariety of a smooth hypersurface of low degree is less than 1.

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arXiv2025

A geometric approach to functional equations for general multiple Dirichlet series over function fields

Sawin recently gave an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb{F}_{q}(T)$ and proved their existence by exhibiting the coefficients as trace functions of specific perverse sheaves. However, he did not prove that these series actually converge anywhere, instead treating them as formal power series. In this paper, we prove that these series do converge in a certain region, and moreover that the functions obtained by analytically continuing them satisfy functional equations. For convergence, it suffices to obtain bounds on the coefficients, for which we use the decomposition theorem for perverse sheaves, in combination with the Kontsevich moduli space of stable maps to construct a suitable compactification. For the functional equations, the key identity is a multi-variable generalization of the relationship between a Dirichlet character and its Fourier transform; in the multiple Dirichlet series setting, this uses a density trick for simple perverse sheaves and an explicit formula for intermediate extensions from the complement of a normal crossings divisor.

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arXiv2018

Sum-Product Phenomena for Planar Hypercomplex Numbers

We study the sum-product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum-product bounds that depend on these parameters. For the dual numbers we expose a range where the minimum value of $\max\{|A+A|,|AA|\}$ is neither close to $|A|$ nor to $|A|^2$. To obtain our main sum-product bound, we extend Elekes' sum-product technique that relies on point-line incidences. Our extension is significantly more involved than the original proof, and in some sense runs the original technique a few times in a bootstrapping manner. We also study point-line incidences in the dual plane and in the double plane, developing analogs of the Szemeredi-Trotter theorem. As in the case of the sum-product problem, it turns out that the dual and double variants behave differently than the complex and real ones.

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arXiv2017

Efficient Point-Counting Algorithms for Superelliptic Curves

In this paper, we present efficient algorithms for computing the number of points and the order of the Jacobian group of a superelliptic curve over finite fields of prime order p. Our method employs the Hasse-Weil bounds in conjunction with the Hasse-Witt matrix for superelliptic curves, whose entries we express in terms of multinomial coefficients. We present a fast algorithm for counting points on specific trinomial superelliptic curves and a slower, more general method for all superelliptic curves. For the first case, we reduce the problem of simplifying the entries of the Hasse-Witt matrix modulo p to a problem of solving quadratic Diophantine equations. For the second case, we extend Bostan et al.'s method for hyperelliptic curves to general superelliptic curves. We believe the methods we describe are asymptotically the most efficient known point-counting algorithms for certain families of trinomial superelliptic curves.

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