Representations of integral quadratic polynomials

Almost universal ternary sums of polygonal numbers

“…For an example of this phenomenon, we direct the reader to [4,Example 4.5]. In our case we are guaranteed that O(L p + ν) contains a symmetry at every prime p. This is obvious at primes p not dividing 2(m − 2) since in this case L p + ν = L p is just a diagonal lattice.…”

“…Following the definitions that originally appear in [4] the class of L + ν is defined as and where O A (V ) denotes the adeles of the kernel of the spinor norm map, θ : SO(V ) → Q × /Q × 2 as defined in [17, §55]. Note that what we refer to as the genus (resp.…”

On a quadratic Waring's problem with congruence conditions

“…is the class number of L (see Corollary 4.4 of [3]). Note that h(L) is equal to the number of elements…”

A characterization of almost universal ternary quadratic polynomials with odd prime power conductor

“…He determined that only seven such triples exist, namely (1, 1, 1), (1,1,2), (1,1,4), (1,1,5), (1,2,2), (1,2,3) and (1,2,4). Recently, in [8], [16], and [18], Z.-W. Sun et al effectively determine all αx 2 + βT y + γT z and αx 2 + βy 2 + γT z that are universal.…”

“…Now an integer n is represented by f if and only if 1 + n is represented by the coset ν + N ; that is, 1 + n = Q(ν + e 1 x + e 2 y + e 3 z)) = (5x + 1) 2 + y 2 + z 2 , which is just a sum of three squares with the added restriction that the first term is congruent to 1 mod 5. For this particular example it is easy to see that f is not almost universal by the three square theorem, but this result will be further confirmed by Theorem 7. When N has rank greater than 3, Chan and Oh [3,Theorem 4.9] show how the asymptotic local-global principles for representations of lattices with approximation property by Jöchner-Kitaoka [12] and by Hsia-Jöchner [11] lead to an asymptotic local-global principle for representations of integers by cosets, consequently we will restrict our attention to ternary N . Imposing some mild arithmetic conditions on f , this paper establishes a characterization of almost universal ternary inhomogeneous quadratic polynomials.…”