@article{cook_hughes_li_mudgal_robert_yung_2024, title={A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem}, volume={70}, ISSN={["2041-7942"]}, url={https://doi.org/10.1112/mtk.12231}, DOI={10.1112/mtk.12231}, abstractNote={Abstract}, number={1}, journal={MATHEMATIKA}, author={Cook, Brian and Hughes, Kevin and Li, Zane Kun and Mudgal, Akshat and Robert, Olivier and Yung, Po-Lam}, year={2024}, month={Jan} }
 @article{dasu_jung_li_madrid_2023, title={Mixed Norm l<SUP>2</SUP> Decoupling for Paraboloids}, volume={2023}, ISSN={["1687-0247"]}, url={https://doi.org/10.1093/imrn/rnad226}, DOI={10.1093/imrn/rnad226}, abstractNote={Abstract}, number={20}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Dasu, Shival and Jung, Hongki and Li, Zane Kun and Madrid, Jose}, year={2023}, month={Oct}, pages={17972–18000} }
 @book{dasu_jung_li_madrid_2023, title={Mixed norm l2 decoupling for paraboloids}, DOI={10.48550/arXiv.2303.04773}, abstractNote={We prove the sharp mixed norm $(l^2, L^{q}_{t}L^{r}_{x})$ decoupling estimate for the paraboloid in $d + 1$ dimensions.}, number={2303.047732303.04773}, author={Dasu, Shival and Jung, Hongki and Li, Zane Kun and Madrid, José}, year={2023} }
 @book{cook_hughes_li_mudgal_robert_yung_2022, title={A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem}, DOI={10.48550/arXiv.2207.01097}, abstractNote={We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely does solution counting in older partial progress on Vinogradov's mean value theorem correspond to in Fourier decoupling theory.}, number={2207.010972207.01097}, author={Cook, Brian and Hughes, Kevin and Li, Zane Kun and Mudgal, Akshat and Robert, Olivier and Yung, Po-Lam}, year={2022} }
 @book{li_2022, title={An introduction to decoupling and harmonic analysis over Qp}, DOI={10.48550/arXiv.2209.01644}, abstractNote={The goal of this expository paper is to provide an introduction to decoupling by working in the simpler setting of decoupling for the parabola over $\mathbb{Q}_p$. Over $\mathbb{Q}_p$, commonly used heuristics in decoupling are significantly easier to make rigorous over $\mathbb{Q}_p$ than over $\mathbb{R}$ and such decoupling theorems over $\mathbb{Q}_p$ are still strong enough to derive interesting number theoretic conclusions.}, number={2209.016442209.01644}, author={Li, Zane Kun}, year={2022} }
 @article{chang_dios pont_greenfeld_jamneshan_li_madrid_2022, title={Decoupling for fractal subsets of the parabola}, volume={301}, ISSN={0025-5874 1432-1823}, url={http://dx.doi.org/10.1007/s00209-021-02950-0}, DOI={10.1007/s00209-021-02950-0}, abstractNote={We consider decoupling for a fractal subset of the parabola. We reduce studying $$l^{2}L^{p}$$ decoupling for a fractal subset on the parabola $$\{(t, t^2) : 0 \le t \le 1\}$$ to studying $$l^{2}L^{p/3}$$ decoupling for the projection of this subset to the interval [0, 1]. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain–Demeter’s decoupling theorem for the parabola. In the case when p/3 is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to [0, 1]. Our ideas are inspired by the recent work on ellipsephic sets by Biggs ( arXiv:1912.04351 , 2019 and Acta Arith. 200(4):331–348, 2021) using nested efficient congruencing.}, number={2}, journal={Mathematische Zeitschrift}, publisher={Springer Science and Business Media LLC}, author={Chang, Alan and Dios Pont, Jaume de and Greenfeld, Rachel and Jamneshan, Asgar and Li, Zane Kun and Madrid, José}, year={2022}, month={Feb}, pages={1851–1879} }
 @article{guo_li_yung_2021, title={A bilinear proof of decoupling for the cubic moment curve}, volume={374}, ISSN={0002-9947 1088-6850}, url={http://dx.doi.org/10.1090/tran/8363}, DOI={10.1090/tran/8363}, abstractNote={Using a bilinear method that is inspired by the method of efficient congruencing of Wooley [Woo16], we prove a sharp decoupling inequality for the moment curve in $\mathbb{R}^3$.}, number={08}, journal={Transactions of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Guo, Shaoming and Li, Zane Kun and Yung, Po-Lam}, year={2021}, month={May}, pages={5405–5432} }
 @article{guo_li_yung_zorin-kranich_2021, title={A short proof of ℓ2 decoupling for the moment curve}, volume={143}, ISSN={1080-6377}, url={http://dx.doi.org/10.1353/ajm.2021.0048}, DOI={10.1353/ajm.2021.0048}, abstractNote={abstract:We give a short and elementary proof of the $\ell^{2}$ decoupling inequality for the moment curve in $\hat{\Bbb{R}}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley.}, number={6}, journal={American Journal of Mathematics}, publisher={Project MUSE}, author={Guo, Shaoming and Li, Zane Kun and Yung, Po-Lam and Zorin-Kranich, Pavel}, year={2021}, pages={1983–1998} }
 @article{li_2021, title={An $l^2$ decoupling interpretation of efficient congruencing: the parabola}, volume={37}, ISSN={0213-2230}, url={http://dx.doi.org/10.4171/rmi/1248}, DOI={10.4171/rmi/1248}, abstractNote={We give a new proof of $l^2$ decoupling for the parabola inspired from efficient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and differences between this new proof and efficient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling such as $l^2 L^2$ decoupling, Bernstein, and ball inflation come into play.}, number={5}, journal={Revista Matemática Iberoamericana}, publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Li, Zane Kun}, year={2021}, month={Feb}, pages={1761–1802} }
 @book{guo_li_yung_2021, title={Improved discrete restriction for the parabola}, DOI={10.48550/arXiv.2103.09795}, abstractNote={Using ideas from Guth-Maldague-Wang and working over $\mathbb{Q}_p$, we show that the discrete restriction constant for the parabola is $O_{\varepsilon}((\log M)^{2 + \varepsilon})$.}, number={2103.097952103.09795}, author={Guo, Shaoming and Li, Zane Kun and Yung, Po-Lam}, year={2021} }
 @article{li_2020, title={EFFECTIVE
            <i>l
            <sup>2
            DECOUPLING FOR THE PARABOLA}, volume={66}, ISSN={0025-5793 2041-7942}, url={http://dx.doi.org/10.1112/mtk.12038}, DOI={10.1112/mtk.12038}, abstractNote={We make effective $l^2 L^p$ decoupling for the parabola in the range $4 < p < 6$. In an appendix joint with Jean Bourgain, we apply the main theorem to prove the conjectural bound for the sixth-order correlation of the integer solutions of the equation $x^2 + y^2 = m$ in an extremal case. This proves unconditionally a result that was proven by Bombieri and Bourgain under the hypotheses of the Birch and Swinnerton-Dyer conjecture and the Riemann Hypothesis for $L$-functions of elliptic curves over $\mathbb{Q}$.}, number={3}, journal={Mathematika}, publisher={Wiley}, author={Li, Zane Kun}, year={2020}, month={May}, pages={681–712} }
 @article{a bilinear proof of decoupling for the cubic moment curve_2019, year={2019}, month={Jun} }
 @article{a short proof of $\ell^2$ decoupling for the moment curve_2019, year={2019}, month={Dec} }
 @phdthesis{li_2019, title={Decoupling for the parabola and connections to efficient congruencing}, url={https://escholarship.org/uc/item/0cz3756c.}, school={University of California Los Angeles}, author={Li, Zane Kun}, year={2019} }
 @article{an $l^2$ decoupling interpretation of efficient congruencing: the parabola_2018, year={2018}, month={May} }
 @article{effective $l^2$ decoupling for the parabola_2017, year={2017}, month={Nov} }
 @article{corwin_feng_li_trebat-leder_2014, title={Elliptic curves with full 2-torsion and maximal adelic Galois representations}, volume={83}, ISSN={0025-5718 1088-6842}, url={http://dx.doi.org/10.1090/s0025-5718-2014-02804-4}, DOI={10.1090/s0025-5718-2014-02804-4}, abstractNote={In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in 

  
    
      
        
          G
          L
        
        2
      
      (
      
        
          
            Z
          
          ^
        
      
      )
    
    \mathrm {GL}_2(\widehat {\mathbb {Z}})
  

. In (2010), Greicius developed necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius’s methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over 

  
    
      
        Q
      
      (
      α
      )
    
    \mathbb {Q}(\alpha )
  

 with maximal image, where 

  
    α
    \alpha
  

 is the real root of 

  
    
      
        x
        3
      
      +
      x
      +
      1
    
    x^3 + x + 1
  

. Next, we extend Greicius’s tests to more general settings, such as non-semistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups.}, number={290}, journal={Mathematics of Computation}, publisher={American Mathematical Society (AMS)}, author={Corwin, David and Feng, Tony and Li, Zane and Trebat-Leder, Sarah}, year={2014}, month={Jan}, pages={2925–2951} }
 @article{li_2014, title={Quadratic twists of elliptic curves with 3-Selmer rank 1}, volume={10}, ISSN={1793-0421 1793-7310}, url={http://dx.doi.org/10.1142/s1793042114500213}, DOI={10.1142/s1793042114500213}, abstractNote={ A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve E/ℚ, a positive proportion of its quadratic twists E(d) have rank 1. Using tools from Galois cohomology, we give criteria on E and d which force a positive proportion of the quadratic twists of E to have 3-Selmer rank 1 and global root number -1. We then give four nonisomorphic infinite families of elliptic curves Em,n which satisfy these criteria. Conditional on the rank part of the Birch and Swinnerton-Dyer conjecture, this verifies the aforementioned conjecture for infinitely many elliptic curves. Our elliptic curves are easy to give explicitly and we state precisely which quadratic twists d to use. Furthermore, our methods have the potential of being generalized to elliptic curves over other number fields. }, number={05}, journal={International Journal of Number Theory}, publisher={World Scientific Pub Co Pte Lt}, author={Li, Zane Kun}, year={2014}, month={Jul}, pages={1191–1217} }
 @article{arithmetic properties of picard-fuchs equations and holonomic recurrences_2013, year={2013}, month={Mar} }
 @article{li_walker_2013, title={Arithmetic properties of Picard–Fuchs equations and holonomic recurrences}, volume={133}, ISSN={0022-314X}, url={http://dx.doi.org/10.1016/j.jnt.2013.02.001}, DOI={10.1016/j.jnt.2013.02.001}, abstractNote={The coefficient series of the holomorphic Picard–Fuchs differential equation associated with the periods of elliptic curves often have surprising number-theoretic properties. These have been widely studied in the case of the torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the Beauville families). Here, we consider arithmetic properties of the Picard–Fuchs solutions associated to general elliptic families, with a particular focus on the index 24 congruence subgroups. We prove that elliptic families with rational parameters admit linear reparametrizations such that their associated Picard–Fuchs solutions lie in Z〚t〛. A sufficient condition is given such that the same holds for holomorphic solutions at infinity. An Atkin–Swinnerton-Dyer congruence is proven for the coefficient series attached to Γ1(7). We conclude with a consideration of asymptotics, wherein it is proved that many coefficient series satisfy asymptotic expressions of the form un∼ℓλn/n. Certain arithmetic results extend to the study of general holonomic recurrences.}, number={8}, journal={Journal of Number Theory}, publisher={Elsevier BV}, author={Li, Zane Kun and Walker, Alexander W.}, year={2013}, month={Aug}, pages={2770–2793} }
 @article{elliptic curves with full 2-torsion and maximal adelic galois representations_2012, year={2012}, month={Jul} }
 @article{a normal form for cubic surfaces_2010, url={http://www.m-hikari.com/ija/ija-2010/ija-5-8-2010/liIJA5-8-2010.pdf}, journal={International Journal of Algebra}, year={2010} }
 @article{li_2010, title={A normal form for cubic surfaces}, volume={4}, number={5}, journal={International Journal of Algebra}, author={Li, Zane Kun}, year={2010}, pages={233–239} }
 @article{li_2010, title={On a special case of the intersection of quadric and cubic surfaces}, volume={214}, ISSN={0022-4049}, url={http://dx.doi.org/10.1016/j.jpaa.2010.02.013}, DOI={10.1016/j.jpaa.2010.02.013}, abstractNote={The intersection curve between two surfaces in three-dimensional real projective space RP3 is important in the study of computer graphics and solid modelling. However, much of the past work has been directed towards the intersection of two quadric surfaces. In this paper we study the intersection curve between a quadric and a cubic surface and its projection onto the plane at infinity. Formulas for the plane and space curves are given for the intersection of a quadric and a cubic surface. A family of cubic surfaces that give the same space curve when we intersect them with a quadric surface is found. By generalizing the methods in Wang et al. (2002) [6] that are used to parametrize the space curve between two quadric surfaces, we give a parametrization for the intersection curve between a quadric and a cubic surface when the intersection has a singularity of order 3.}, number={11}, journal={Journal of Pure and Applied Algebra}, publisher={Elsevier BV}, author={Li, Zane Kun}, year={2010}, month={Nov}, pages={2078–2086} }
 @article{humphries_li_2009, title={Counting powers of words in monoids}, volume={30}, ISSN={0195-6698}, url={http://dx.doi.org/10.1016/j.ejc.2008.10.005}, DOI={10.1016/j.ejc.2008.10.005}, abstractNote={For a monoid M with presentation M = 〈 a 1 , … , a r | w 1 = w 2 , … , w 2 s − 1 = w 2 s 〉 , we count the number of words equivalent to w 1 n , n ∈ N , where equivalent means under the transitive closure of the relation generated by replacing an occurrence of w 2 i − 1 by w 2 i or vice versa (for any i ). Many interesting sequences are obtained in this way including the Fibonacci numbers.}, number={5}, journal={European Journal of Combinatorics}, publisher={Elsevier BV}, author={Humphries, Stephen P. and Li, Zane Kun}, year={2009}, month={Jul}, pages={1297–1308} }