Combinatorial Completions

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

An integer may be represented by a quadratic form over each ring of p -adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

An integral quadratic form is called strictly n-regular if it primitively represents all quadratic forms in n variables that are primitively represented by its genus. For any $$n \ge 2$$ , it will be shown that there are only finitely many similarity classes of positive definite strictly n-regular integral quadratic forms in $$n + 4$$ variables. This extends the recent finiteness results for strictly regular quaternary quadratic forms by Earnest et al. (J Number Theory 144: 256–266, 2014).

In this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.

In this note we prove an elementary result concerning the distribution of the positive integers which are represented by a positive-definite integral binary quadratic form Theorem 1. Let f(X, Y) = aX ~ + b X Y + c y2 be a positive-definite integral binary quadratic form of discriminant - A (= b 2 - 4ac = mi , there exist integers x and y such that n < ax 2 q- bxy + cy 2 < n + 2ml/~A1/4n ~m + m i . P r 0 0 f. Replacing the form f by an equivalent form we may suppose that m~ = c. We define integers x and y by =F(4cn~l/2q I!4cn-Ax2)l/2-bx l x L\~) /' Y= 2c -+1.

For each positive integer $n$, let $g_\Delta (n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd squares. In t

A treatment by integral matrices is given for composition of pairs of integral quadratic forms with the same discriminant. The forms are associated with a pair of similar 2 × 2 matrices A B with irrational eigen values which generate the maximal order. The most general integral similarity between A B is given by a matrix whose entries are linear forms in two indeterminates with integral coefficients. This matrix is a compositum of two factors of the same nature. By equating determinants a composition of two quadratic forms results. The method can be generalized to n × n matrices.

Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares of $m$ integers? Denoting this set of all possible $T$ by $\mathscr{S}_m(n)$, Goldmakher and Pollack have given a simple characterization of $\mathscr{S}_4(n)$ using elementary arguments. Their result can be reinterpreted in terms of Mordell's theory of representations of binary integral quadratic forms as sums of squares of integral linear forms. Based on this approach, we characterize $\mathscr{S}_m(n)$ for all $m\leq 11$ and provide a few partial results for arbitrary $m$. We also show how Mordell's results can be used to study variations of the original problem where the sum of the integers is replaced by a linear form in these integers. In this way, we recover and generalize earlier results by Z.W. Sun et. al..

Refining the notion of regularity introduced by Dickson, an integral quadratic form is said to be spinor regular if it represents all integers represented by its spinor genus. Examples of positive definite primitive integral ternary quadratic forms which have this property are presented, and it is proved that there exist only finitely many equivalence classes containing such forms.

There are close relations between Bernoulli numbers and quadratic fields or quadratic forms. In order to explain those, we give a survey of the ideal theory of quadratic fields and quadratic forms. Since Gauss , it is well known that there is a deep relation between the ideal theory of quadratic fields (i.e. quadratic extensions of the rational number field) and integral quadratic forms. This is obvious for specialists, but textbooks which explain this in detail are rare. In most textbooks, they treat the correspondence of ideals only in the case of maximal orders (the ring of all integers) for which the description is simple and easy. This is very disappointing. We cannot see the whole picture of the theory of quadratic forms by such a restricted treatment and sometimes it causes misunderstanding. We need the full description of this kind of relation when we explain later the relation between L-functions of prehomogeneous vector spaces and the Bernoulli numbers. So it would be a good chance to try to explain the full relation here.

SynopsisA formula is given for the number of genera of primitive integral binary quadratic forms of discriminant D which lie in a rational equivalence class. In particular, necessary and sufficient conditions for the genus and rational equivalence class to coincide are given in terms of the prime factorisation of D.

SynopsisWe compute the number of ℚ-classes of primitive integral binary quadratic forms of any discriminant and characterise when it coincides with the number of genera.

Small representations of integers by integral quadratic forms

Latin squares are combinatorial constructions that have found widespread application in communication systems through frequency hopping designs, error correcting codes and encryption algorithms. In this paper, a new, lower bound on the cardinality of the critical sets of all Latin squares of order n is shown to be Ω(n!(n-3)!) where Ω is the summatory prime factorisation function (with multiplicities). The proof draws on the direct product of the symmetric groups S n and S n-3 . The smallest critical set cardinalities of the new bound align with its known, calculated values and reduce previously proven bounds for n > 8. The proof refutes the long standing Nelder conjecture.

Root systems and inflations of non-negative quasi-Cartan matrices