2,527 research outputs found
An elementary proof of Hilbert's theorem on ternary quartics
In 1888, Hilbert proved that every non-negative quartic form f=f(x,y,z) with
real coefficients is a sum of three squares of quadratic forms. His proof was
ahead of its time and used advanced methods from topology and algebraic
geometry. Up to now, no elementary proof is known. Here we present a completely
new approach. Although our proof is not easy, it uses only elementary
techniques. As a by-product, it gives information on the number of
representations f=p_1^2+p_2^2+p_3^2 of f up to orthogonal equivalence. We show
that this number is 8 for generically chosen f, and that it is 4 when f is
chosen generically with a real zero. Although these facts were known, there was
no elementary approach to them so far.Comment: 26 page
On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring
Chevyrev and Galbraith recently devised an algorithm which inputs a maximal
order of the quaternion algebra ramified at one prime and infinity and
constructs a supersingular elliptic curve whose endomorphism ring is precisely
this maximal order. They proved that their algorithm is correct whenever it
halts, but did not show that it always terminates. They did however prove that
the algorithm halts under a reasonable assumption which they conjectured to be
true. It is the purpose of this paper to verify their conjecture and in turn
prove that their algorithm always halts.
More precisely, Chevyrev and Galbraith investigated the theta series
associated with the norm maps from primitive elements of two maximal orders.
They conjectured that if one of these theta series "dominated" the other in the
sense that the th (Fourier) coefficient of one was always larger than or
equal to the th coefficient of the other, then the maximal orders are
actually the same. We prove that this is the case.Comment: 12 page
The representation of integers by positive ternary quadratic polynomials
An integral quadratic polynomial is called regular if it represents every
integer that is represented by the polynomial itself over the reals and over
the -adic integers for every prime . It is called complete if it is of
the form , where is an integral quadratic
form in the variables and is a
vector in . Its conductor is defined to be the smallest positive
integer such that . We prove that for a
fixed positive integer , there are only finitely many equivalence classes of
positive primitive ternary regular complete quadratic polynomials with
conductor . This generalizes the analogous finiteness results for positive
definite regular ternary quadratic forms by Watson and for ternary triangular
forms by Chan and Oh
Some new canonical forms for polynomials
We give some new canonical representations for forms over \cc. For example,
a general binary quartic form can be written as the square of a quadratic form
plus the fourth power of a linear form. A general cubic form in
can be written uniquely as a sum of the cubes of linear forms
, . A general ternary quartic form
is the sum of the square of a quadratic form and three fourth powers of linear
forms. The methods are classical and elementary.Comment: I have spoken about this material under the title "steampunk
canonical forms". This is the final revised version which has been accepted
by the Pacific Journal of Mathematics. Apart from the usual improvements
which come after a thoughtful refereeing, Theorem 1.8 is ne
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