Quadratic Waring's Problem

• Quadratic Waring’s Problem is a framework investigating how quadratic forms can be uniformly represented as sums of squares in integers and other rings.
• Modern methods replace exponential-in-n bounds with sub-exponential-in-√n bounds, leveraging balanced HKZ reduction and lattice-neighbor arguments to refine classical invariant estimates.
• Extensions to Hermitian forms and congruence conditions reveal intricate connections between additive combinatorics, density conditions, and symmetric tensor rank in quadratic settings.

Quadratic Waring’s problem denotes a family of representation problems centered on quadratic expressions. In its classical additive form, one asks for the least number of squares needed so that every sufficiently large integer is representable as a sum of that many integer squares. In arithmetic theory of quadratic forms, the phrase usually refers to a higher-dimensional analogue: given an integral quadratic form in $n$ variables that is already known to be a sum of squares of integral linear forms, determine a uniform bound on how many such squares are actually necessary. Modern work extends this perspective to Hermitian forms over imaginary quadratic fields, quadratic forms over totally real number fields, congruence-constrained representations by odd squares, density versions over cyclic groups and primes, and related function-field and tensor-decomposition settings (Beli et al., 2017, Chan et al., 2020, Kim, 2019, Lim, 20 Aug 2025).

1. Core formulations and invariants

The classical quadratic Waring problem over $\mathbb Z$ asks, for fixed $s$, whether every sufficiently large integer $y$ can be written as

$y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$

Lagrange’s theorem settles the extremal case $s=4$: every nonnegative integer is a sum of four squares (Lim, 20 Aug 2025).

The higher-dimensional form-theoretic version replaces integers by quadratic forms. For a commutative ring $R$, an $r$-ary quadratic form $q\in R[X_1,\dots,X_r]$ is called admissible if it is represented by a sum of squares of linear forms,

$q(X)=L_1(X)^2+\cdots+L_N(X)^2$

for some $\mathbb Z$0. Writing $\mathbb Z$1 for the set of such forms, the invariant $\mathbb Z$2 is the minimal $\mathbb Z$3 such that every $\mathbb Z$4 is already a sum of $\mathbb Z$5 squares of linear forms (Krásenský et al., 2021).

For the integral case $\mathbb Z$6, one may equivalently define $\mathbb Z$7 as the smallest $\mathbb Z$8 such that every positive-definite integral quadratic form in $\mathbb Z$9 variables, once known to be represented by some sum of squares of integral linear forms, is represented by the standard sum-of-$s$0-squares form

$s$1

(Beli et al., 2017).

The number-field and Hermitian variants replace $s$2 by rings of integers. If $s$3 is totally real with ring of integers $s$4, then $s$5 is the smallest integer $s$6 such that every sum of squares of $s$7-ary $s$8-linear forms is itself a sum of $s$9 such squares (Chan et al., 2020). If $y$0 is an imaginary quadratic field of class number one with ring of integers $y$1, then $y$2 is defined analogously for positive-definite integral Hermitian forms represented by a sum of norms

$y$3

(Beli et al., 2017).

A related lattice-theoretic refinement is the invariant $y$4, defined as the least $y$5 such that the standard lattice $y$6 represents every $y$7-ary quadratic lattice $y$8 over $y$9 with $y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$0 for some $y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$1. One has

$y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$2

(Krásenský et al., 2021).

Invariant	Setting	Meaning
$y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$3	Integral quadratic forms	Uniform square-count for $y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$4-ary forms representable by sums of squares
$y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$5	Totally real number fields	Same problem over $y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$6
$y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$7	Hermitian forms over imaginary quadratic fields	Uniform norm-count
$y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$8	Quadratic lattices	Lattice version of the representation invariant
$y=x_1^2+\cdots+x_s^2,\qquad x_i\in\mathbb Z.$9	Complete quadratic polynomials	Uniform odd-square count under congruence conditions

At $s=4$0, $s=4$1 is the usual Pythagoras number $s=4$2, the smallest number of squares needed to represent every sum of squares in $s=4$3 (Krásenský et al., 2021).

2. Sub-exponential bounds for quadratic-form invariants

The central asymptotic development in the arithmetic theory is the replacement of exponential-in-$s=4$4 bounds by exponential-in-$s=4$5 bounds. For $s=4$6 or an imaginary quadratic field of class number one with ring of integers $s=4$7, define the Euclidean minimum

$s=4$8

and

$s=4$9

Then for every $R$0,

$R$1

In particular, for $R$2 one has $R$3 and

$R$4

(Beli et al., 2017).

This improves earlier bounds of Conway–Sloane and Kim–Oh of the form

$R$5

with Kim–Oh obtaining roughly $R$6 (Beli et al., 2017). The improvement is asymptotically substantial: the governing scale drops from $R$7 in the exponent to $R$8.

The totally real number-field version was established by extending HKZ reduction from $R$9 to totally real fields. If $r$0 is totally real of class number $r$1 with ring of integers $r$2, then there exist constants $r$3 and $r$4, depending only on $r$5, such that

$r$6

for all $r$7 (Chan et al., 2020). This is the first sub-exponential upper bound for $r$8 when $r$9 (Chan et al., 2020).

A further extension removes the class-number-one restriction. For a totally real number field $q\in R[X_1,\dots,X_r]$0 of degree $q\in R[X_1,\dots,X_r]$1,

$q\in R[X_1,\dots,X_r]$2

and more generally for any subfield $q\in R[X_1,\dots,X_r]$3,

$q\in R[X_1,\dots,X_r]$4

Consequently,

$q\in R[X_1,\dots,X_r]$5

and for each $q\in R[X_1,\dots,X_r]$6 there is $q\in R[X_1,\dots,X_r]$7 such that

$q\in R[X_1,\dots,X_r]$8

(Krásenský et al., 2021).

The exact growth remains unknown. Mordell’s original analogy suggests a linear law $q\in R[X_1,\dots,X_r]$9, but already $q(X)=L_1(X)^2+\cdots+L_N(X)^2$0; current lower bounds are essentially linear in $q(X)=L_1(X)^2+\cdots+L_N(X)^2$1, whereas the best known upper bounds are $q(X)=L_1(X)^2+\cdots+L_N(X)^2$2 (Chan et al., 2020).

3. Reduction theory and proof architecture

The modern upper bounds rely on a reduction-theoretic framework built around balanced HKZ reduction. In the Hermitian and integral setting, a positive-definite form $q(X)=L_1(X)^2+\cdots+L_N(X)^2$3 in $q(X)=L_1(X)^2+\cdots+L_N(X)^2$4 variables can be integrally transformed into a balanced HKZ-reduced form whose Gram matrix factors as

$q(X)=L_1(X)^2+\cdots+L_N(X)^2$5

where $q(X)=L_1(X)^2+\cdots+L_N(X)^2$6 and $q(X)=L_1(X)^2+\cdots+L_N(X)^2$7 has both $q(X)=L_1(X)^2+\cdots+L_N(X)^2$8 and $q(X)=L_1(X)^2+\cdots+L_N(X)^2$9 with entries bounded by $\mathbb Z$00; in addition the diagonal coefficients satisfy

$\mathbb Z$01

(Beli et al., 2017).

Over a totally real field $\mathbb Z$02, the corresponding statement writes the Gram matrix as

$\mathbb Z$03

with $\mathbb Z$04 upper-triangular unipotent, $\mathbb Z$05, and $\mathbb Z$06. Balanced HKZ reduction provides a nondecreasing function $\mathbb Z$07, a function $\mathbb Z$08, and a constant $\mathbb Z$09 depending only on $\mathbb Z$10 such that for $\mathbb Z$11 and every infinite place $\mathbb Z$12,

$\mathbb Z$13

and

$\mathbb Z$14

(Chan et al., 2020).

These inequalities control the successive minima and support a decomposition of the form

$\mathbb Z$15

or

$\mathbb Z$16

where $\mathbb Z$17 is diagonal and chosen as large as possible while preserving positive semidefiniteness, $\mathbb Z$18 is manifestly a sum of squares, and $\mathbb Z$19 is a small symmetric error term whose off-diagonal entries satisfy bounds of the shape

$\mathbb Z$20

with $\mathbb Z$21 (Beli et al., 2017, Chan et al., 2020).

Two mechanisms then complete the argument. First, a lattice-neighbor argument generalizing Kneser–Schiemann shows that $\mathbb Z$22 is represented by a controlled number of squares; in the totally real case, a technical lemma yields representation by

$\mathbb Z$23

squares when the diagonal entries are sufficiently large (Chan et al., 2020). Second, an induction on minima or on rank removes the large-minimum assumption. In the Hermitian setting this gives

$\mathbb Z$24

while in the totally real case one obtains a recursion

$\mathbb Z$25

(Beli et al., 2017, Chan et al., 2020).

A distinct but complementary technique appears in the extension-of-scalars approach. If an order $\mathbb Z$26 has degree $\mathbb Z$27 over $\mathbb Z$28, then representing an $\mathbb Z$29-ary form over $\mathbb Z$30 by $\mathbb Z$31 produces an associated $\mathbb Z$32-submodule of rank at most $\mathbb Z$33, leading to

$\mathbb Z$34

The lattice version satisfies

$\mathbb Z$35

for $\mathbb Z$36 an $\mathbb Z$37-module of rank $\mathbb Z$38, and this bypasses any class-number-one hypothesis (Krásenský et al., 2021).

4. Lattice invariants and congruence-constrained variants

The lattice invariant $\mathbb Z$39 makes it possible to ask for uniform representation of all $\mathbb Z$40-ary quadratic lattices that already embed into some $\mathbb Z$41. This formulation is especially effective over totally real number fields. One application determines $\mathbb Z$42 for almost all real quadratic fields: if $\mathbb Z$43 is not one of $\mathbb Z$44, $\mathbb Z$45, or $\mathbb Z$46, then

$\mathbb Z$47

For the exceptional fields, known or expected values are

$\mathbb Z$48

while for $\mathbb Z$49 one expects $\mathbb Z$50, pending a suitable local–global principle for integral binary forms (Krásenský et al., 2021).

A separate variant imposes congruence conditions on the summands. Let

$\mathbb Z$51

and let

$\mathbb Z$52

be a complete quadratic polynomial. Writing $\mathbb Z$53 for the set of such $\mathbb Z$54-variable polynomials representable by some $\mathbb Z$55, define

$\mathbb Z$56

and then

$\mathbb Z$57

Thus $\mathbb Z$58 is the least $\mathbb Z$59 such that every representable complete quadratic polynomial in $\mathbb Z$60 variables is representable by at most $\mathbb Z$61 odd squares (Kim, 2019).

This invariant has the same qualitative asymptotic growth as the classical quadratic-form invariant: $\mathbb Z$62 More explicitly, the paper obtains

$\mathbb Z$63

(Kim, 2019).

The small-rank values are exact: $\mathbb Z$64 so $\mathbb Z$65 for $\mathbb Z$66 (Kim, 2019). These differ sharply from the classical integral invariant, for which $\mathbb Z$67 is known for $\mathbb Z$68, and $\mathbb Z$69 (Kim, 2019).

5. Additive, density, and prime-square forms

In additive combinatorics, the “density version” of the quadratic Waring problem asks when a dense subset of the quadratic residues mod $\mathbb Z$70 remains additively universal. If

$\mathbb Z$71

and $\mathbb Z$72 has relative density $\mathbb Z$73, then $\mathbb Z$74 is called $\mathbb Z$75-representable mod $\mathbb Z$76 if every residue class is a sum of $\mathbb Z$77 elements of $\mathbb Z$78. For $\mathbb Z$79 and $\mathbb Z$80, there exists $\mathbb Z$81 such that whenever

$\mathbb Z$82

and $\mathbb Z$83 satisfies $\mathbb Z$84, the set $\mathbb Z$85 is $\mathbb Z$86-representable mod $\mathbb Z$87. Moreover, for every $\mathbb Z$88,

$\mathbb Z$89

The threshold $\mathbb Z$90 is sharp: for every $\mathbb Z$91 and all sufficiently large primes $\mathbb Z$92, there exists $\mathbb Z$93 with $\mathbb Z$94 but $\mathbb Z$95 (Lim, 20 Aug 2025).

The corresponding quadratic Waring–Goldbach density problem replaces squares by squares of primes. For $\mathbb Z$96, define

$\mathbb Z$97

If $\mathbb Z$98 primes has lower relative density

$\mathbb Z$99

then every sufficiently large integer $s$00 is representable as

$s$01

For example, when $s$02, the threshold is $s$03, improving the previous $s$04 (Lim, 20 Aug 2025).

A more specialized recent development concerns Piatetski–Shapiro primes. For any $s$05, every sufficiently large integer $s$06 can be written as

$s$07

with each

$s$08

for some $s$09. The proof uses a transference principle rather than a full Hardy–Littlewood asymptotic formula, and concludes existence of at least one representation for all large admissible $s$10 (Gao et al., 28 Feb 2026).

These density and prime-square formulations are not identical to the invariant $s$11, but they share the same structural theme: one seeks uniform representation from a prescribed quadratic family under quantitative sparsity or density constraints.

6. Other settings and current directions

The quadratic Waring problem also appears over function fields. For $s$12 with $s$13, define $s$14 as the number of representations

$s$15

with each $s$16 of degree $s$17. Then for all $s$18,

$s$19

uniformly for all large admissible $s$20, and consequently

$s$21

In this quadratic case the minor-arc set is empty, so the full circle-method analysis is major-arc in nature (Yamagishi, 2015).

A different terminological branch belongs to algebraic geometry and symmetric tensor rank. There one studies Waring decompositions of powers of a quadratic form

$s$22

The apolar ideal satisfies

$s$23

where $s$24 is the space of harmonic polynomials of degree $s$25, and the rank obeys

$s$26

For $s$27,

$s$28

except in the tight cases $s$29; for $s$30, $s$31; and for $s$32, the minimal ranks are $s$33, $s$34, and $s$35 for $s$36, respectively (Flavi, 2024). This is a distinct problem from the arithmetic theory of integral quadratic forms, although the shared vocabulary reflects a common concern with decomposing quadratic objects into powers of linear ones.

Several open problems remain central. The exact asymptotic growth of $s$37 and $s$38 is unknown; the existing gap between essentially linear lower bounds and $s$39 upper bounds remains wide (Chan et al., 2020). Over totally real fields, further refinement of balanced HKZ bounds could lower the constants in the exponent, and extensions to rings of higher class number or to Hermitian forms over CM fields remain natural targets (Chan et al., 2020). In the congruence-constrained setting, $s$40 is still unresolved, with the available evidence leaving $s$41 or $s$42 as plausible values (Kim, 2019). For the Piatetski–Shapiro prime-square problem, an asymptotic formula with singular series and power-saving error remains open by current methods (Gao et al., 28 Feb 2026).