This thesis consists of two parts. In Part I we present a new class of norm discretization inequalities suited for low-degree polynomials in many dimensions, with applications to discrete harmonic analysis and to quantum and classical learning theory.
\r\n\r\nDiscretization inequalities (of Bernstein type) control the supremum norm of polynomials f by their supremum norms over certain finite subsets T of the domain. Unlike earlier multivariate Bernstein-type discretization inequalities we establish dimension-free comparisons for simple and generic T, such as product sets T=S\u2081 \u00d7 \u22c5\u22c5\u22c5 \u00d7 S\u2099 for S\u2c7c's consisting of well-spread points in R or C, in exchange for a constant that grows with deg(f).
\r\n\r\nOur results also introduce the notion of \"individual degree\"\u2014the maximum degree of f in any one variable\u2014as a fundamental parameter for discretization inequalities: we show for the first time that dimension-free discretizations of the uniform norm are possible for T with cardinality independent of deg(f), provided f has bounded individual degree.
\r\n\r\nOur work offers a new, high-dimensional perspective on discretization inequalities and yields several new results in analysis on the hypergrid (i.e., products of cyclic groups), including Bohnenblust\u2013Hille-type inequalities, dimension-free supremum norm bounds on level-k Fourier projections, and junta theorems. These estimates in turn provide the key analytic tools for extending recent breakthroughs in learning low-degree functions to the hypergrid and to its quantum analogue, local observables on K-level qudit systems.
\r\n\r\nIn Part II we apply ideas from analysis of Boolean functions to study other aspects of (quantum) computation: circuit complexity and property testing.
\r\n\r\nFirst, we introduce and study a deceptively simple model of constant-depth quantum circuits and begin the project of proving bounds on its capabilities, ultimately drawing on connections to nonlocal games and notions of approximate degree.
\r\n\r\nSecond, we introduce a new access model for property testing, quantum data, which allows for ultrafast testing algorithms where classical data provably yields no fast testers\u2014such as for monotonicity, symmetry, and triangle-freeness.
", "doi": "10.7907/grjv-rz74", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:10241", "collection": "thesis", "collection_id": "10241", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012017-013622968", "primary_object_url": { "basename": "Guo_Zeyu_2017.pdf", "content": "final", "filesize": 1548403, "license": "other", "mime_type": "application/pdf", "url": "/10241/1/Guo_Zeyu_2017.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "P-Schemes and Deterministic Polynomial Factoring Over Finite Fields", "author": [ { "family_name": "Guo", "given_name": "Zeyu", "orcid": "0000-0001-7893-4346", "clpid": "Guo-Zeyu" } ], "thesis_advisor": [ { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C-M" } ], "thesis_committee": [ { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C-M" }, { "family_name": "Schulman", "given_name": "Leonard J.", "orcid": "0000-0001-9901-2797", "clpid": "Schulman-L-J" }, { "family_name": "Vidick", "given_name": "Thomas G.", "orcid": "0000-0002-6405-365X", "clpid": "Vidick-T" }, { "family_name": "Huang", "given_name": "Ming-Deh", "clpid": "Huang-Ming-Deh" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "We introduce a family of mathematical objects called P-schemes, where P is a poset of subgroups of a finite group G. A P-scheme is a collection of partitions of the right coset spaces H\\G, indexed by H\u2208P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes [BI84] as well as the notion of m-schemes [IKS09].
\r\n\r\nBased on P-schemes, we develop a unifying framework for the problem of deterministic factoring of univariate polynomials over finite field under the generalized Riemann hypothesis (GRH). More specifically, our results include the following:
\r\n\r\nWe show an equivalence between m-scheme as introduced in [IKS09] and P-schemes in the special setting that G is an multiply transitive permutation group and P is a poset of pointwise stabilizers, and therefore realize the theory of m-schemes as part of the richer theory of P-schemes.
\r\n\r\nWe give a generic deterministic algorithm that computes the factorization of the input polynomial \u0192(X) \u2208 Fq[X] given a \"lifted polynomial\" \u0192~(X) of \u0192(X) and a collection F of \"effectively constructible\" subfields of the splitting field of \u0192~(X) over a certain base field. It is routine to compute \u0192~(X) from \u0192(X) by lifting the coefficients of \u0192(X) to a number ring. The algorithm then successfully factorizes \u0192(X) under GRH in time polynomial in the size of \u0192~(X) and F, provided that a certain condition concerning P-schemes is satisfied, for P being the poset of subgroups of the Galois group G of \u0192~(X) defined by F via the Galois correspondence. By considering various choices of G, P and verifying the condition, we are able to derive the main results of known (GRH-based) deterministic factoring algorithms [Hua91a; Hua91b; Ron88; Ron92; Evd92; Evd94; IKS09] from our generic algorithm in a uniform way.
\r\n \r\nWe investigate the schemes conjecture in [IKS09] and formulate analogous conjectures associated with various families of permutation groups, each of which has applications on deterministic polynomial factoring. Using a technique called induction of P-schemes, we establish reductions among these conjectures and show that they form a hierarchy of relaxations of the original schemes conjecture.
\r\n\r\nWe connect the complexity of deterministic polynomial factoring with the complexity of the Galois group G of \u0192~(X). Specifically, using techniques from permutation group theory, we obtain a (GRH-based) deterministic factoring algorithm whose running time is bounded in terms of the noncyclic composition factors of G. In particular, this algorithm runs in polynomial time if G is in \u0393k for some k=2O(\u221a(log n), where \u0393k denotes the family of finite groups whose noncyclic composition factors are all isomorphic of subgroups of the symmetric group of degree k. Previously, polynomial-time algorithms for \u0393k were known only for bounded k.
\r\n\r\nWe discuss various aspects of the theory of P-schemes, including techniques of constructing new P-schemes from old ones, P-schemes for symmetric groups and linear groups, orbit P-schemes, etc. For the closely related theory of m-schemes, we provide explicit constructions of strongly antisymmetric homogeneous m-schemes for m\u22643. We also show that all antisymmetric homogeneous orbit 3-schemes have a matching for m\u22653, improving a result in [IKS09] that confirms the same statement for m\u22654.
\r\n\r\nIn summary, our framework reduces the algorithmic problem of deterministic polynomial factoring over finite fields to a combinatorial problem concerning P-schemes, allowing us to not only recover most of the known results but also discover new ones. We believe progress in understanding P-schemes associated with various families of permutation groups will shed some light on the ultimate goal of solving deterministic polynomial factoring over finite fields in polynomial time.
", "doi": "10.7907/Z94F1NSG", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" } ]