1 Introduction and main results
A quadratic form over the ring of integers
$\mathcal {O}_K$
of a totally real number field
$K$
is universal if it represents every totally positive integer. Universal quadratic forms have a long history over
$\mathbb {Z}$
, beginning with Lagrange’s theorem that
$x_1^2+x_2^2+x_3^2+x_4^2$
represents every positive integer and culminating in the
$290$
-theorem of Bhargava and Hanke [Reference Bhargava and Hanke1], by which a positive definite integral quadratic form is universal as soon as it represents an explicit finite list of integers. Over a totally real field
$K$
, one replaces positivity by total positivity and asks for forms representing every totally positive element of
$\mathcal {O}_K$
. Already, sums of squares behave subtly in this setting: Siegel [Reference Siegel10] proved that
$x_1^2+\cdots +x_n^2$
is universal over
$\mathcal {O}_K$
only for
$K=\mathbb {Q}$
and
$K=\mathbb {Q}(\sqrt 5)$
.
Over real quadratic fields
$\mathbb {Q}(\sqrt D)$
, the minimal number of variables in a universal form and related invariants are controlled by the continued fraction expansion of
$\sqrt D$
and by the totally positive indecomposable integers, those totally positive elements of
$\mathcal {O}_K$
that cannot be written as a sum of two totally positive integers (see Blomer and Kala [Reference Blomer and Kala2] and the survey [Reference Kala7]). Generalised quadratic forms, introduced over totally real fields by Browning et al. [Reference Browning, Pierce and Schindler3], enlarge the class of quadratic forms by allowing the Galois conjugates of a variable to occur alongside the variable and encode the arithmetic of systems of quadrics. Their universality was studied by the Prague school [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Reference Chwiedziuk, Doležálek, Pěchoučková, Pezlar, Prakash, Romeo, Růžičková and Zindulka5].
In [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4], Chwiedziuk et al. proved the following important dichotomy over real quadratic fields. Call a generalised form proper if it has a variable
$z$
with both
$z$
and its conjugate present. Then, under a positive-definiteness hypothesis, a universal generalised form must contain a universal ordinary quadratic subform; the definiteness hypothesis is necessary, with a semidefinite counterexample over
$\mathbb {Q}(\sqrt 2)$
[Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Theorem 7, Example 9]. That proof is special to degree two. As noted in [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Remark 8], its engine is a lower bound on the symmetric expression
$z^2+\tau (z)^2=\mathrm {Tr}(z^2)$
, available because a degree two field has exactly two conjugates, so a proper variable carries both. In degree
$d\geqslant 3$
, a proper variable may carry only two of
$d$
conjugates, with no symmetric expression to bound from below, and on this basis, the authors conjecture that a counterexample should exist over a cyclic cubic field.
We settle this. The phenomenon they isolate is real; a partial proper variable can be made small in one embedding, but it yields no counterexample, because it cannot be made small in two embeddings at once. This is the content of our main result, proved in Section 3 as Theorem 3.1 and sharpening [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Lemma 6]: if a totally positive definite generalised form represents a value through a nonzero proper variable, then that value is bounded below in at least two of the
$d$
Archimedean embeddings. A proper variable can therefore contribute to representing a totally positive integer only when that integer is large in at least two embeddings, and from this, the universality theorem is immediate, in full generality.
Theorem 1.1 (Universality).
Let
$K$
be any totally real number field of degree
${d\geqslant 2}$
and let
$G$
be a totally positive definite generalised quadratic form over
$\mathcal {O}_K$
built from
$H=\mathrm {Aut}(K/\mathbb {Q})$
. If
$G$
is universal, then
$G$
contains a universal quadratic subform. Consequently,
$G$
is universal if and only if its quadratic subform is so.
In particular, the higher-degree counterexample conjectured in [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Remark 8] does not exist. Theorem 1.1 removes both the degree restriction and any Galois assumption; for
$K$
Galois of degree
$d$
, one has
$|H|=d$
, recovering the setting of Remark 8.
Theorem 1.1 follows from the estimate of Section 3 by one observation: given
$\alpha \in \mathcal {O}_K^{+}$
, a unit
$\varepsilon $
can be chosen so that
$\varepsilon ^2\alpha $
is small in any prescribed
$d-1$
embeddings simultaneously, possible because the unit group of a totally real field of degree
$d$
has rank
$d-1$
. Thus, the totally real hypothesis serves twice, supplying real embeddings for definiteness and the unit rank
$d-1$
for the shrinking. Beyond Dirichlet’s unit theorem, the argument only requires that a nonzero algebraic integer cannot have all Archimedean absolute values below
$1$
.
This bound is sharp. Over the smallest cyclic cubic field
$\mathbb {Q}(\zeta _7)^{+}$
, we exhibit in Section 5 a definite proper form
$B(z)=z^2+\sigma (z)^2$
whose value is large in two embeddings and tends to
$0$
in the third, so the count two cannot be improved to three for
$d=3$
and the single-embedding argument of [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4] fails verbatim for
$d\geqslant 3$
. A companion witness for a partial proper variable over the non-Galois quartic field
$\mathbb {Q}(\sqrt {3+\sqrt 2})$
is given in Example 4.4.
Finally, total positive-definiteness cannot be relaxed in degree two: [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Example 9] constructs a totally positive semidefinite universal generalised form over
$\mathbb {Q}(\sqrt 2)$
with no universal quadratic subform. Whether the same failure occurs in degree
$d\geqslant 3$
is the natural remaining question and we record it as an open problem in Section 6.
2 The setup
Let
$K$
be a totally real number field of degree
$d$
, with ring of integers
$\mathcal {O}_K$
, distinct real embeddings
$\sigma _1,\dots ,\sigma _d\colon K\hookrightarrow \mathbb {R}$
and automorphism group
$H=\mathrm {Aut}(K/\mathbb {Q})$
. For
$\alpha \in K$
, write
$\mathrm {N}(\alpha )=\prod _k\sigma _k(\alpha )$
and
$\mathrm {Tr}(\alpha )=\sum _k\sigma _k(\alpha )$
; call
$\alpha $
totally positive,
$\alpha \gg 0$
, if
$\sigma _k(\alpha )>0$
for all
$k$
, and let
$\mathcal {O}_K^{+}$
be the totally positive integers and
$\mathcal {O}_K^{\times }$
the units in
$\mathcal {O}_K$
.
2.1 Generalised quadratic forms
We follow [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Definitions 1–3] in the degree
$d$
formulation of [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Remark 8]. To each variable
$z_i$
(
$1\leqslant i\leqslant r$
), attach a nonempty support
$P_i\subseteq H$
with a formal variable
$w_{i,\rho }$
for each
$\rho \in P_i$
. A generalised quadratic form is
where
$Q$
, the associated form, is an ordinary quadratic form with coefficients in
$\mathcal {O}_K$
(that is,
$G$
is integral). A variable
$z_i$
is cosmetic if
$|P_i|=1$
and proper if
$|P_i|\geqslant 2$
; a proper variable is full if
$P_i=H$
. The quadratic subform is
$\widetilde {Q}:=G|_{z_i=0\ \text {for proper }i}$
, an ordinary quadratic form in the cosmetic variables. As in [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Definition 2],
$G$
is totally positive definite if
$\sigma _k(Q)$
and the real form, obtained by applying
$\sigma _k$
to the coefficients, is positive definite for every
$k$
; equivalently,
$Q\gg 0$
on
$K^{n}\setminus \{0\}$
. It is universal if it represents every
$\alpha \in \mathcal {O}_K^{+}$
. Since each
$\rho \in H$
is a bijection of
$\mathcal {O}_K$
, the cosmetic conjugations do not affect which integers are represented, so
$\widetilde {Q}$
is universal as a generalised form if and only if the underlying ordinary form is so [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Section 2].
2.2 Two lemmas
The first is [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Lemma 5] and is the only use of definiteness.
Lemma 2.1. If
$G$
is totally positive definite with associated form
$Q$
in
$n$
variables, there is a rational
$\delta =\delta (K,G)>0$
with
$\sigma _k(Q)(w_1,\ldots ,w_n)\geqslant \delta (w_1^2+\cdots +w_n^2)$
for every embedding
$\sigma _k$
and all real
$w$
.
Proof. For each
$k$
, take
$\delta _k>0$
to be the least eigenvalue of the positive-definite form
$\sigma _k(Q)$
, so
$\sigma _k(Q)(w)\geqslant \delta _k\|w\|^2$
, and set
$\delta =\min _k\delta _k$
.
The second is where the degree
$d$
enters; in degree two, it is [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Lemma 4], shrinking the single nondistinguished embedding. We deduce it from the statement of Dirichlet’s unit theorem.
Lemma 2.2. Let
$K$
be totally real of degree
$d$
and fix any
$d-1$
of its embeddings. For every
$C>0$
, there is a unit
$\varepsilon \in \mathcal {O}_K^{\times }$
with
$|\sigma _k(\varepsilon )|<C$
for each of the chosen
$d-1$
embeddings.
Proof. By Dirichlet’s unit theorem [Reference Neukirch9, Ch. I, Section 7], [Reference Marcus8, Ch. 5], the logarithmic embedding
$\mathrm {Log}\colon \mathcal {O}_K^{\times }\to \mathbb {R}^{d}$
,
$\eta \mapsto (\log |\sigma _1\eta |,\dots ,\log |\sigma _d\eta |)$
, maps
$\mathcal {O}_K^{\times }$
onto a full lattice in the trace-zero hyperplane
$H_0=\{x\in \mathbb {R}^{d}:\sum _k x_k=0\}$
. Projection of
$H_0$
onto any
$d-1$
of the coordinates is a linear isomorphism onto
$\mathbb {R}^{d-1}$
, so this lattice contains a point all of whose chosen
$d-1$
coordinates lie below
$\log C$
; any unit mapping to such a point satisfies the conclusion.
We use two further facts. If
$z\in \mathcal {O}_K$
has
$|\sigma _k(z)|<1$
for all
$k$
, then
$|\mathrm {N}(z)|=\prod _k|\sigma _k(z)|<1$
and as
$\mathrm {N}(z)\in \mathbb {Z}$
, this forces
$z=0$
. Further,
$H$
acts freely on the embeddings by
$\sigma \mapsto \sigma \circ \rho $
: if
$\sigma \circ \rho =\sigma $
, then injectivity of
$\sigma $
gives
$\rho =\mathrm {id}$
, so for distinct
$\rho ,\rho '\in H$
, the embeddings
$\sigma \circ \rho ,\sigma \circ \rho '$
are distinct.
3 The two-embedding bound
Theorem 3.1. Let
$K$
be a totally real number field of degree
$d$
, let
$G$
be a totally positive definite generalised quadratic form over
$\mathcal {O}_K$
and let
$\delta>0$
be a definiteness constant for its associated form (Lemma 2.1). If
$z_1,\dots ,z_r\in \mathcal {O}_K$
are such that some proper variable
$z_i$
takes a nonzero value, then
If
$z_i$
is full, meaning all
$d$
conjugates of
$z_i$
occur, then
$\sigma _k(G(\mathbf z))\geqslant \delta \,\mathrm {Tr}(z_i^2)\geqslant \delta d$
for every
$k$
.
Proof. Let
$\delta $
be as in Lemma 2.1 and let
$z_i$
be a proper variable with
$z_i\ne 0$
. For each embedding
$\sigma _k$
, writing the associated variables as
$w_{i',\rho '}=\rho '(z_{i'})$
and discarding all nonnegative terms but those attached to
$z_i$
,
Suppose, for contradiction, that
$\sigma _k(G(\mathbf z))\geqslant \delta $
for at most one index
$k$
; then, at least
${d-1}$
embeddings have
$\sigma _k(G(\mathbf z))<\delta $
. Choose
$d-1$
of them and relabel as
$\sigma _1,\dots ,\sigma _{d-1}$
, leaving
$\sigma _d$
as the only possibly excluded embedding. By (3.1), for
$k\leqslant d-1$
,
Fix
$m\in \{1,\dots ,d\}$
and pick distinct
$\rho ,\rho '\in P_i$
, possible since
$|P_i|\geqslant 2$
. By freeness,
${\sigma _m\circ \rho ^{-1}}$
and
$\sigma _m\circ \rho ^{\prime -1}$
are distinct, so not both equal
$\sigma _d$
; choose
$\rho "\in \{\rho ,\rho '\}$
with
$\sigma _m\circ \rho ^{\prime \prime -1}=\sigma _k$
,
$k\leqslant d-1$
. Then,
$\sigma _m=\sigma _k\circ \rho "$
gives
$|\sigma _m(z_i)|=|\sigma _k(\rho "(z_i))|<1$
. As
$m$
was arbitrary, all
$d$
absolute values of
$z_i$
lie below
$1$
, forcing
$z_i=0$
, which is a contradiction. Hence, at least two embeddings satisfy
$\sigma _k(G(\mathbf z))\geqslant \delta $
.
If the variable
$z_i$
is full, then for each
$k$
, the multiset
$\{\sigma _k(\rho (z_i)):\rho \in H\}$
is equal to
$\{\sigma _m(z_i):1\leqslant m\leqslant d\}$
by simple transitivity, so
$\sum _{\rho \in H}\sigma _k(\rho (z_i))^2=\mathrm {Tr}(z_i^2)$
, and (3.1) gives
$\sigma _k(G(\mathbf z))\geqslant \delta \,\mathrm {Tr}(z_i^2)$
. Finally,
$z_i^2\in \mathcal {O}_K^{+}\setminus \{0\}$
has
$\mathrm {N}(z_i^2)\geqslant 1$
, so by the arithmetic-geometric mean inequality,
$\mathrm {Tr}(z_i^2)\geqslant d\,\mathrm {N}(z_i^2)^{1/d}\geqslant d$
.
4 Universality and an equivalence
Proof of Theorem 1.1.
Let
$\delta $
be the constant of Lemma 2.1 and
$\alpha \in \mathcal {O}_K^{+}$
arbitrary. Put
$M=1+\max _k\sigma _k(\alpha )$
, choose
$C>0$
with
$C^2M<\delta $
, and by Lemma 2.2, take a unit
$\varepsilon $
with
$|\sigma _k(\varepsilon )|<C$
for
$k\leqslant d-1$
. Set
$\beta :=\varepsilon ^2\alpha \in \mathcal {O}_K$
. Then,
$\beta \gg 0$
and
$\sigma _k(\beta )=\sigma _k(\varepsilon )^2\sigma _k(\alpha )<C^2M<\delta $
for
$k\leqslant d-1$
, so at most one embedding has
$\sigma _k(\beta )\geqslant \delta $
. As
$G$
is universal and
$\beta \gg 0$
, write
$\beta =G(z_1,\dots ,z_r)$
. By Theorem 3.1, no proper variable can be nonzero, since that would force two embeddings of
$\beta $
to be
$\geqslant \delta $
. Hence,
$\beta =\widetilde {Q}(\mathbf y)$
for cosmetic values
$\mathbf y\in \mathcal {O}_K^{s}$
, and as
$\widetilde {Q}$
is an ordinary quadratic form,
$\widetilde {Q}(\varepsilon ^{-1}\mathbf y)=\varepsilon ^{-2}\beta =\alpha $
with
$\varepsilon ^{-1}\mathbf y\in \mathcal {O}_K^{s}$
; the cosmetic conjugations being bijections of
$\mathcal {O}_K$
, this is a representation of
$\alpha $
by
$\widetilde {Q}$
. So,
$\widetilde {Q}$
is universal. The converse is trivial, since
$\widetilde {Q}$
arises from
$G$
by setting variables to zero, so every value of
$\widetilde {Q}$
is a value of
$G$
.
Corollary 4.1. Let
$G$
be totally positive definite over a totally real field
$K$
, with constant
$\delta $
. If
$\beta \in \mathcal {O}_K^{+}$
is represented by
$G$
and
$\sigma _k(\beta )<\delta $
for at least
$d-1$
embeddings, then
$\beta $
is represented by the quadratic subform
$\widetilde {Q}$
. Equivalently, proper variables can enlarge the represented set only by integers that are
$\geqslant \delta $
in at least two embeddings.
Proof. Write
$\beta =G(\mathbf z)$
. By Theorem 3.1, no proper variable is nonzero, so
$\beta =\widetilde {Q}(\mathbf y)$
.
Remark 4.2. In degree two, ‘at least two embeddings’ means both, so Theorem 3.1 recovers and slightly strengthens [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Lemma 6], which records only the distinguished embedding. The strengthening is invisible there, as in degree two, every proper variable is full and full variables are controlled in all embeddings. More generally, the full case of Theorem 3.1 gives
$\sigma _k(G(\mathbf z))\geqslant \delta \,\mathrm {Tr}(z_i^2)$
in every embedding, the exact analogue of the symmetric bound of the quadratic case. For a partial proper variable, the trace stays large, but can concentrate in
$d-1$
embeddings, and Theorem 3.1 shows it can never concentrate to leave fewer than two.
Remark 4.3 (Non-Galois fields).
The proof of Theorem 3.1 uses only the facts that
$H=\mathrm {Aut}(K/\mathbb {Q})$
acts freely on the embeddings and that proper means
$|P_i|\geqslant 2$
; transitivity is never used. Hence, Theorems 3.1 and 1.1 hold over any totally real field with the generalised form notion built from
$\mathrm {Aut}(K/\mathbb {Q})$
. When
$\mathrm {Aut}(K/\mathbb {Q})$
is trivial, as for a non-Galois cubic, there are no proper variables and the statements are vacuous. The first nontrivial non-Galois case is a quartic with
$\mathrm {Aut}(K/\mathbb {Q})\cong C_2$
, treated in the following example.
Example 4.4 (A non-Galois quartic).
Let
$K=\mathbb {Q}(t)$
with
$t=\sqrt {3+\sqrt 2}$
, the field of
$x^4-6x^2+7$
. Its four roots
$\pm \sqrt {3\pm \sqrt 2}$
are real, so
$K$
is totally real and, by the standard criterion for biquadratic quartics, with
$p=-6$
,
$q=7$
, neither
$q=7$
nor
${q(p^2-4q)=56}$
being a square, the Galois closure is
$D_4$
. Thus,
$K$
is non-Galois with
$\mathrm {Aut}(K/\mathbb {Q})=\{1,\tau \}\cong C_2$
, where
$\tau (t)=-t$
, and the unique quadratic subfield is
$\mathbb {Q}(\sqrt 2)=\mathbb {Q}(t^2-3)$
. The involution
$\tau $
acts on the four embeddings
$\sigma _k\colon t\mapsto \pm \sqrt {3\pm \sqrt 2}$
without fixed points, pairing
$\{\sigma _1,\sigma _2\}$
and
$\{\sigma _3,\sigma _4\}$
.
The proper form
$B(z)=z^2+\tau (z)^2$
has associated form
$u^2+v^2$
, positive definite, so
$B$
is totally positive definite with
$\delta =1$
and uses two of the four conjugates. Theorem 3.1 predicts that at least two of the four embeddings of
$B(z)$
are
$\geqslant 1$
for every nonzero
$z$
and the covering step reads concretely here: among the two distinct embeddings
$\sigma _m,\sigma _m\circ \tau $
, at least one survives the exclusion of any single embedding, so controlling three embeddings forces
$|\sigma _m(z)|<1$
for all
$m$
; hence,
$z=0$
. The bound two is attained: for
$z=2-t^2$
, the four values of
$\sigma _k(B(z))$
are
$11.66\ldots ,\,11.66\ldots ,\,0.343\ldots ,\,0.343\ldots $
, large in exactly two embeddings. A search over
$z=a+bt+ct^2+dt^3$
with
$|a|,|b|,|c|,|d|\leqslant 3$
confirms that the number of embeddings with
$\sigma _k(B(z))\geqslant 1$
is never below two.
5 Sharpness over
$\mathbb {Q}(\zeta _7)^{+}$
Let
$K=\mathbb {Q}(\zeta _7)^{+}=\mathbb {Q}(\theta )$
with
$\theta =2\cos (2\pi /7)$
the smallest cyclic cubic field. Here,
$\theta $
has minimal polynomial
$x^3+x^2-2x-1$
, the ring of integers is
$\mathbb {Z}[\theta ]$
and the generator
$\sigma $
of
$\mathrm {Gal}(K/\mathbb {Q})\cong C_3$
acts by
$\sigma (\theta )=\theta ^2-2$
. Consider the proper form
which uses the two conjugates
$z,\sigma (z)$
and not
$\sigma ^2(z)$
, with associated form
$u^2+v^2$
positive definite. Thus,
$B$
is totally positive definite, with
$\delta =1$
admissible, and
$B(z)\gg 0$
for
$z\ne 0$
.
Proposition 5.1. Theorem 3.1 is sharp: the count two cannot be replaced by three for
$d=3$
. With
$z=227+328\,\theta -409\,\theta ^2$
, one has
$B(z)=2417977-1247022\,\theta -554977\,\theta ^2\in \mathcal {O}_K^{+}$
and
so exactly two embeddings are
$\geqslant \delta $
. Moreover,
Proof. The displayed values are computed in
$\mathbb {Z}[\theta ]$
and verified to
$40$
digits. For the infimum,
$\sigma _1(B(z))=\sigma _1(z)^2+\sigma _2(z)^2$
; with
$z=u^n$
for the unit
$u=-2-2\theta +3\theta ^2$
, whose embeddings are
$(0.1709\ldots ,-0.5157\ldots ,11.34\ldots )$
, both
$|\sigma _1(u)|,|\sigma _2(u)|<1$
, so
$\sigma _1(B(u^n))=\sigma _1(u)^{2n}+\sigma _2(u)^{2n}\to 0$
while
$B(u^n)\gg 0$
throughout.
This is the configuration anticipated in [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Remark 8]: a definite proper variable carrying two of three conjugates makes
$\sigma _1(B(z))$
arbitrarily small, so the single-embedding argument of [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Lemma 6, Theorem 7] fails verbatim for
${d\geqslant 3}$
. What survives is that
$B(z)$
cannot be small in two embeddings at once, since
$\sigma _1(B(z)),\sigma _2(B(z))<1$
would force
$\sigma _1(z)^2,\sigma _2(z)^2,\sigma _3(z)^2<1$
; hence,
$|\mathrm {N}(z)|<1$
and
$z=0$
. Shrinking the target in
$d-1=2$
embeddings, as in Theorem 1.1, exploits exactly this and is available because the unit rank of a cubic field is
$2$
.
6 An open problem
Total positive-definiteness cannot be relaxed in Theorem 1.1, at least in degree two. Over
$\mathbb {Q}(\sqrt 2)$
, [Reference Chwiedziuk, Doležálek, Fryšová, Pěchoučková, Pezlar, Prakash, Růžičková and Zindulka4, Example 9] builds a semidefinite universal generalised form
${G=S+H}$
with no universal quadratic subform. By a theorem of Cohn [Reference Cohn6], the sum of four squares
$S$
represents all totally positive elements of the index-two sublattice
$\mathbb {Z}[2\sqrt 2]$
of
$\mathbb {Z}[\sqrt 2]$
, while
$H$
, a sum of suitably weighted squares
$\ell (z)^2$
of the proper linear form
$\ell (z)=2z-\tau (z)$
, supplies the missing coset. The squares
$\ell (z)^2$
are rank one, so
$G$
is only semidefinite and every quadratic subform of
$G$
is a subform of
$S$
; hence, not universal. In our argument, definiteness is used exactly once, in Lemma 2.1; for a merely semidefinite form, the constant
$\delta $
is
$0$
and Theorem 3.1 collapses, in any degree. This leaves the following question.
Problem 6.1. Let
$K$
be a totally real number field of degree
$d\geqslant 3$
with
${\mathrm {Aut}(K/\mathbb {Q})\neq \{1\}}$
so that proper variables exist over
$K$
. Does there exist a totally positive semidefinite generalised quadratic form over
$\mathcal {O}_K$
that is universal but has no universal quadratic subform? More generally, determine the totally real fields over which such a form exists.
Acknowledgements
The authors thank the Exeter22 team, advisors and investors for making this research possible. The authors also thank Emily Nie, Diamante Gruosi, Miles Chapman and Johnny Ni for their helpful suggestions on the proof of the Main Theorem during the preparation of this manuscript. The authors thank the referee for valuable comments on earlier drafts of this paper.
