Search

For n, d ∈ ℕ, the cone 퓟n+1,2d of positive semidefinite real forms in n + 1 variables of degree 2d contains the subcone Σn+1,2d of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the Hilbert cases (n + 1, 2d) with n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4). In this paper, we induce a filtration of intermediate cones between Σn+1,2d and 퓟n+1,2d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties Vk−n ⊊ ... ⊊ Vn ⊊ ... ⊊ V0 containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n + 1 variables of degree d. By showing that V0, ..., Vn (and Vn+1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σn+1,2d. We moreover prove that, in the non-Hilbert cases of (n + 1)-ary quartics for n ≥ 3 and (n + 1)-ary sextics for n ≥ 2, all the remaining cone inclusions are strict. [ABSTRACT FROM AUTHOR]

We determine all possible values of the integer group determinant of C 2 4 , where C 2 is the cyclic group of order 2. [ABSTRACT FROM AUTHOR]

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all x ∈ R n if and only if there exists a sum-of-squares (SOS) polynomial σ (x) such that σ P is a sum of sparse SOS matrices. Second, we show that setting σ (x) = (x 1 2 + ⋯ + x n 2) ν for some integer ν suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set K = { x : g 1 (x) ≥ 0 , ... , g m (x) ≥ 0 } satisfying the Archimedean condition, then P (x) = S 0 (x) + g 1 (x) S 1 (x) + ⋯ + g m (x) S m (x) for matrices S i (x) that are sums of sparse SOS matrices. Finally, if K is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for (x 1 2 + ⋯ + x n 2) ν P (x) with some integer ν ≥ 0 when P and g 1 , ... , g m are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones. [ABSTRACT FROM AUTHOR]

We give a necessary and sufficient condition for symmetric tensors to have symmetric rank decompositions and give some further results on the Comon's Conjecture (i.e. the rank and symmetric rank agree for a symmetric tensor). In particular, as an application, we prove that if an m-order 2-dimensional symmetric tensor has a symmetric rank decomposition, then its symmetric rank equals its rank or its rank plus one. [ABSTRACT FROM AUTHOR]

For a set of integers A, we consider R (A) = { a / b : a , b ∈ A , b ≠ 0 } . It is an open problem to study the denseness of R(A) in the p-adic numbers when A is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms a x 3 + b y 3 , where a and b are integers. In this article, we study this problem for diagonal forms. We extend their results to the diagonal binary forms a x n + b y n for all n ≥ 3 . We also study p-adic denseness of quotients of nonzero values attained by diagonal forms of degree n ≥ 3 , where gcd (n , p (p - 1)) = 1 . [ABSTRACT FROM AUTHOR]

Bhargava parametrized quintic rings over Z by quadruples of 5 × 5 alternating matrices. We extend the construction to work similarly over any Dedekind domain R. No assumptions are needed on the characteristic of R. The resolvent consists of a pair of locally free modules L, M with two multilinear maps between them; we can view L as Q/R, for Q the quintic ring, and M as S/R, where S is a sextic resolvent ring. As in Bhargava's treatment, any quintic ring has a resolvent ring, and for a maximal ring, the resolvent is unique. We hope that this work will enable the removal of the condition that the characteristic be different from 2 in Bhargava-Shankar-Wang's proof of Linnik's conjecture on the asymptotic distribution of discriminants of relative extensions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over ${\mathbb {Q}}$ satisfying $\max \{|a_2|, |a_1|, |a_0|\} \leq X$ is bounded from above by $O(X (\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava–Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves. [ABSTRACT FROM AUTHOR]

We investigate the existence of representations of every large positive integer as a sum of k-th powers of integers represented as certain diagonal forms. In particular, we consider a family of diagonal forms and discuss the problem of giving a uniform upper bound over the family for the number of variables needed to have such representations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence of nontrivial, nonsingular zeros of a nonhomogeneous polynomial defined over a finite field. To accomplish this, we determine conditions that guarantee the existence of a prescribed number of nonsingular zeros of a homogeneous form f over a finite field k that are not zeros of a homogeneous form h when f, h are relatively prime. The cases of quadratic and cubic polynomials are considered in detail. This extends previous results that have usually considered only the homogeneous case. [ABSTRACT FROM AUTHOR]

For A ⊆ { 1 , 2 , ... } , we consider R (A) = { a / b : a , b ∈ A } . It is an open problem to study the denseness of R(A) in the p-adic numbers when A is the set of nonzero values assumed by a cubic form. We study this problem for the cubic forms a x 3 + b y 3 , where a and b are integers. We also prove that if A is the set of nonzero values assumed by a non-degenerate, integral, and primitive cubic form with more than 9 variables, then R(A) is dense in Q p . [ABSTRACT FROM AUTHOR]

Let d and n be positive integers and let K be a totally real number field of discriminant d and degree n. We construct a theta series θ K ∈ M d , n , where M d , n is a space of modular forms defined in terms of n and d. Moreover, if d is square free and n is at most 4 then θ K is a complete invariant for K. We also investigate whether or not the collection of θ -series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant d, is a linearly independent subset of M d , 4 . This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well. [ABSTRACT FROM AUTHOR]

Let λ π (1 , ... , 1 , n) be the normalized Fourier coefficients of an even Hecke–Maass form π for S L (m , Z) with m ≥ 3 , and r 3 (n) = # { (n 1 , n 2 , n 3) ∈ Z 3 : n = n 1 2 + n 2 2 + n 3 2 } . In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of GL(m) Fourier coefficients λ π (1 , ... , 1 , n) and r 3 (n) , which improves previous results (even under the generalized Ramanujan conjecture). [ABSTRACT FROM AUTHOR]

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$ , having good reduction at $2$ and $3$ , and whose $j$ -invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient $\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$ -Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. The key new ingredients necessary for the proofs are 'uniformity estimates', namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes. [ABSTRACT FROM AUTHOR]

We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley. We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point. [ABSTRACT FROM AUTHOR]

Let F be a non-singular homogeneous polynomial of degree d in n variables. We give an asymptotic formula of the pairs of integer points (x , y) with | x | ⩽ X and | y | ⩽ Y which generate a line lying in the hypersurface defined by F, provided that n > 2 d - 1 d 4 (d + 1) (d + 2) . In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of X and Y. [ABSTRACT FROM AUTHOR]

We prove that a system of R diagonal equations of degree d over a finite extension K of Qp has a non‐trivial solution in K if the number of variables exceeds 3R2d2 (if p>2) or 8R2d2 (if p=2). As a consequence, a system of R homogeneous equations of degree d over K has a non‐trivial solution in K if the number of variables exceeds (8R2d2)2d−1. [ABSTRACT FROM AUTHOR]

In this paper, we study the shifted convolution sums of the Fourier coefficients λ π (1 , ... , 1 , n) and r s , k (n) with k ≥ 3 , where r s , k (n) denotes the number of representations of the positive integer n as sums of skth powers. We are able to generalize or improve previous results. [ABSTRACT FROM AUTHOR]

Here we study the recently introduced notion of a locally harmonic Maass form and its applications to the theory of L‐functions. In particular, we find a criterion for vanishing of certain twisted central L‐values of a family of elliptic curves, whereby vanishing occurs precisely when the values of two finite sums over canonical binary quadratic forms coincide. This yields a vanishing criterion based on vastly simpler formulas than the formulas related to work of Birch and Swinnerton‐Dyer to determine such L‐values, and extends beyond their framework to special non‐CM elliptic curves. [ABSTRACT FROM AUTHOR]

The elliptic curve Ek:y2=x3+k admits a natural 3‐isogeny ϕk:Ek→E−27k. We compute the average size of the ϕk‐Selmer group as k varies over the integers. Unlike previous results of Bhargava and Shankar on n‐Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on k; this sensitivity can be precisely controlled by the Tamagawa numbers of Ek and E−27k. As a consequence, we prove that the average rank of the curves Ek, k∈Z, is less than 1.21 and over 23% (respectively, 41%) of the curves in this family have rank 0 (respectively, 3‐Selmer rank 1). [ABSTRACT FROM AUTHOR]

Let F be a binary form with integer coefficients, non-zero discriminant and degree d with d at least 3. Let R F (Z) denote the number of integers of absolute value at most Z which are represented by F. We prove that there is a positive number C F such that R F (Z) is asymptotic to C F Z 2 d . [ABSTRACT FROM AUTHOR]

In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings h ( Δ ) of a given discriminant Δ, over the integers, to the number of cubic rings h ˆ ( Δ ) of discriminant − 27 Δ in which every element has trace divisible by 3: (1) h ˆ ( Δ ) = { 3 h ( Δ ) if Δ > 0 h ( Δ ) if Δ < 0 , where in each case, rings are weighted by the reciprocal of their number of automorphisms. This allows the functional equations governing the analytic continuation of the Shintani zeta functions (the Dirichlet series built from the functions h and h ˆ ) to be put in self-reflective form. In 1998, J. Nakagawa verified (1) . We present a new proof of (1) that uses the main ingredients of Nakagawa's proof (binary cubic forms, recursions, and class field theory), as well as one of Bhargava's celebrated higher composition laws, while aiming to stay true to the stark elegance of the identity. [ABSTRACT FROM AUTHOR]

The K -rank of a binary form f in K [ x , y ] , K ⊆ C , is the smallest number of d -th powers of linear forms over K of which f is a K -linear combination. We provide lower bounds for the C -rank (Waring rank) and for the R -rank (real Waring rank) of binary forms depending on their factorization. We study binary forms with unique C -minimal representation. [ABSTRACT FROM AUTHOR]

Hilbert proved in 1888 that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if n = 2 or d = 1 or ( n , 2 d ) = ( 3 , 4 ) , where n is the number of variables and 2 d the degree of the form. We study the analogue for even symmetric forms. We establish that an even symmetric n -ary 2 d -ic psd form is sos if and only if n = 2 or d = 1 or ( n , 2 d ) = ( n , 4 ) n ≥ 3 or ( n , 2 d ) = ( 3 , 8 ) . [ABSTRACT FROM AUTHOR]

We establish that smooth, geometrically integral projective varieties of small degree are not pointless in suitable solvable extensions of their field of definition, provided that this field has characteristic zero. [ABSTRACT FROM AUTHOR]

A famous theorem of Hilbert from 1888 states that for given n and d , every positive semidefinite (psd) real form of degree 2 d in n variables is a sum of squares (sos) of real forms if and only if n = 2 or d = 1 or ( n , 2 d ) = ( 3 , 4 ) . In 1976, Choi and Lam proved the analogue of Hilbert's Theorem for symmetric forms by assuming the existence of psd not sos symmetric n -ary quartics for n ≥ 5 . In this paper we complete their proof by constructing explicit psd not sos symmetric n -ary quartics for n ≥ 5 . [ABSTRACT FROM AUTHOR]

The aim of this paper is to find and prove generalizations of some of the beautiful integral parametrizations in Bhargava's theory of higher composition laws to the case where the base ring $$\mathbb {Z}$$ is replaced by an arbitrary Dedekind domain R. Specifically, we parametrize quadratic, cubic, and quartic algebras over R as well as ideal classes in quadratic algebras, getting a description of the multiplication law on ideals that extends Bhargava's famous reinterpretation of Gauss composition of binary quadratic forms. We expect that our results will shed light on the statistical properties of number field extensions of degrees 2, 3, and 4. [ABSTRACT FROM AUTHOR]

We establish the Hasse principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6 r and the associated coefficient matrix contains no singular $$r\times r$$ submatrix, thereby achieving the theoretical limit of the circle method for such systems. [ABSTRACT FROM AUTHOR]

We show that Artin's conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r ≥ 2. [ABSTRACT FROM AUTHOR]