Humans have been thinking about polynomial equations over the integers, or over the rational numbers, for many years. Despite this, their secrets are tightly locked up and it is hard to know what to expect, even in simple looking cases. In this talk I’ll discuss recent efforts to understand the frequency of integer solutions to cubic polynomials, before turning to the much more evolved picture over the rational numbers.
How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in [-H, H], is O(H3.91). More generally, we show that if n ≥ 3 and n ≠ 7, 8, 10 then there are O(Hn-1.017) monic, irreducible polynomials of degree n with integer coefficients in [-H, H] and Galois group not containing An. Save for the alternating group and degrees 7, 8, 10, this establishes a 1936 conjecture of van der Waerden, that irreducible non-Sn polynomials are substantially rarer than reducible polynomials.
Let us fix a prime p and a homogeneous system of m linear equations aj,1x1+ . . . +aj,kxk=0 for j=1, . . ., m with coefficients aj,i ∈ 𝔽p. Suppose that k ≥ 3m, that a_{j,1}+ . . . +a_{j,k}=0 for j=1,. . . ,m and that every m × m minor of the m × k matrix (a_{j,i})_{j,i} is non-singular. Then we prove that for any (large) n, any subset A ⊆ (𝔽p)n of size |A| greater than C Γn contains a solution (x1, . . .,xk) ∈ Ak to the given system of equations such that the vectors x1, . . .,xk ∈ A are all distinct. Here, C and Γ are constants only depending on p, m and k such that Γ < p. The crucial point here is the condition for the vectors x1, . . .,xk in the solution (x1, . . .,xk) ∈ Ak to be distinct. If we relax this condition and only demand that x1, . . .,xk are not all equal, then the statement would follow easily from Tao's slicerank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slicerank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.
Dual functions, known in ergodic theory as multiple correlation sequences, are an important but poorly-understood class of functions in additive combinatorics. An example of such a function is one that, given a subset A and element d, counts the number of arithmetic progressions in A with common difference d. To make progress on an equally poorly understood probabilistic version of Szemerédi's theorem with random common differences, it has been suggested to determine if dual functions can be decomposed in terms of "higher-order characters" (polynomial phases or nilsequences) plus a small error function. Conjectured bounds for Szemerédi's theorem with random differences were motivated by an apparent expectation that the error can always be taken to have small L∞ norm. It turns out that this is too much to hope for. In this talk we discuss counterexamples to such decompositions, ideas of which originate from coding theory.
Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.
Suppose for each prime p we are given a set Ap (possibly empty) of residue classes mod p. Use these and the Chinese Remainder Theorem to form a set Aq of residue classes mod q, for any integer q. Under very mild hypotheses, we show that for a typical integer q, the residue classes in Aq will become equidistributed. The prototypical example (which this generalises) is Hooley's theorem that the roots of a polynomial congruence mod n are equidistributed on average over n. I will also discuss generalisations of such results to higher dimensions, and when restricted to integers with a given number of prime factors. (Joint work with Emmanuel Kowalski.)
I’ll survey some of the key challenges around the solubility of polynomial Diophantine equations over the integers. While studying individual equations is often extraordinarily difficult, the situation is more accessible if we merely ask what happens on average and if we restrict to the so-called Fano range, where the number of variables exceeds the degree of the polynomial. Indeed, about 20 years ago, it was conjectured by Poonen and Voloch that random Fano hypersurfaces satisfy the Hasse principle, which is the simplest necessary condition for solubility. After discussing related results I’ll report on joint work with Pierre Le Boudec and Will Sawin where we establish this conjecture for all Fano hypersurfaces, except cubic surfaces.
