Macaulay Matrix Method Overview

• Macaulay Matrix Method is a linear algebraic framework that converts nonlinear polynomial and differential relations into sparse, monomial-indexed linear systems.
• It is applied in diverse contexts including SOS programming, quantum polynomial solving, and Feynman integral computations, demonstrating flexibility across algebraic settings.
• The approach leverages combinatorial constructions to analyze sparsity, rank bounds, and condition numbers, offering practical linearizations for complex algebraic problems.

Searching arXiv for relevant papers on the Macaulay Matrix Method to ground the article in published work. Querying arXiv for papers explicitly involving Macaulay matrices, including algebraic, SOS, quantum, and Feynman-integral contexts. The Macaulay Matrix Method is a linear-algebraic framework in which polynomial multiplication, ideal membership, prolongation, or differential-operator action is encoded in a matrix indexed by monomials or derivative monomials. In recent arXiv literature, the term appears in several technically distinct realizations: the classical Macaulay matrix and Macaulay linear system for multivariate polynomial solving over $\mathbb{C}$, the Macaulay representation of the prolongation matrix $J_{n,d}$ for diagonal Hermitian sums of squares, and a differential-operator Macaulay matrix in the rational Weyl algebra for deriving Pfaffian systems for GKZ and Feynman integrals (Ding et al., 2021, Wang et al., 4 Sep 2025, Chestnov et al., 2022).

1. Scope and structural forms

The common feature across these realizations is the replacement of nonlinear algebraic relations by finite-dimensional linear algebra on monomially indexed coordinates. In the classical polynomial-system setting, the matrix records coefficients of products $m f$ for monomials $m$ and input polynomials $f$. In the SOS setting, the matrix records multiplication by $S_1(x)=x_1+\cdots+x_n$ on coefficient vectors of diagonal Hermitian forms. In the GKZ/Feynman setting, the matrix records normal forms of $\partial^k\circ h_j$ in a quotient of a Weyl algebra.

Context	Indexing pattern	Immediate role
Polynomial systems over $\mathbb{C}$	Rows $(m,f)$, columns monomials of degree at most $d$	Defines the Macaulay linear system $J_{n,d}$0
Diagonal Hermitian SOS	Degree-$J_{n,d}$1 and degree-$J_{n,d}$2 monomial bases in $J_{n,d}$3	Tests $J_{n,d}$4 and counts rank via support
GKZ and Feynman integrals	Rows from $J_{n,d}$5, columns derivative monomials	Solves for Pfaffian matrices and linear relations

The cited papers therefore use a shared Macaulay principle rather than a single universal matrix. What is preserved is the passage from multiplicative structure to sparse linear constraints, together with a monomial basis adapted to the application (Ding et al., 2021, Wang et al., 4 Sep 2025, Chestnov et al., 2022).

2. Classical Macaulay matrices and polynomial-system linearization

For $J_{n,d}$6 variables $J_{n,d}$7 and degree parameter $J_{n,d}$8, the standard monomial count is

$J_{n,d}$9

Given $m f$0, the Macaulay matrix of degree $m f$1, denoted $m f$2, has rows indexed by pairs $m f$3 where $m f$4 and $m f$5 is a monomial such that $m f$6 has degree at most $m f$7, and columns indexed by monomials of degree at most $m f$8. The entry in row $m f$9 and column $m$0 is the coefficient of $m$1 in $m$2. Writing the augmented matrix as

$m$3

one obtains the Macaulay linear system

$m$4

In the Boolean setting over $m$5, the construction used in the quantum-algorithm literature first embeds the problem into $m$6 by adding field equations $m$7. Chen and Gao’s approach, as analyzed in the later paper, also introduces auxiliary integers $m$8 and replaces them by binary variables $m$9 through

$f$0

For a Boolean assignment $f$1, the corresponding monomial evaluation vector has entries

$f$2

If $f$3 is the Hamming weight of $f$4, then for max degree $f$5 the number of nonzero coordinates is $f$6, while for total degree $f$7 it is $f$8. Accordingly,

$f$9

These norms drive the condition-number analysis for HHL-type quantum linear-system algorithms. With $S_1(x)=x_1+\cdots+x_n$0, the paper proves that if the system has $S_1(x)=x_1+\cdots+x_n$1 solutions of minimum Hamming weight $S_1(x)=x_1+\cdots+x_n$2, and either all solutions have the same Hamming weight or the minimum-$S_1(x)=x_1+\cdots+x_n$3 solution lies in the convex hull of the solution vectors, then

$S_1(x)=x_1+\cdots+x_n$4

for max-degree Macaulay matrices and

$S_1(x)=x_1+\cdots+x_n$5

for total-degree Macaulay matrices. In Chen–Gao’s choice $S_1(x)=x_1+\cdots+x_n$6, this yields $S_1(x)=x_1+\cdots+x_n$7, which underlies the comparison showing that Grover search is at least as fast as the original Macaulay-plus-HHL approach in the unique-solution regime considered there (Ding et al., 2021).

3. Macaulay representation of the prolongation matrix in Hermitian SOS theory

In the SOS setting, let $S_1(x)=x_1+\cdots+x_n$8 and let $S_1(x)=x_1+\cdots+x_n$9 be a real-valued diagonal bihomogeneous Hermitian polynomial. With a left lexicographic basis of holomorphic monomials up to degree $\partial^k\circ h_j$0,

$\partial^k\circ h_j$1

there exists a Hermitian matrix $\partial^k\circ h_j$2 such that

$\partial^k\circ h_j$3

For diagonal Hermitian polynomials, $\partial^k\circ h_j$4 is diagonal in this basis, and $\partial^k\circ h_j$5 is a sum of squares precisely when $\partial^k\circ h_j$6.

The paper passes to real variables by setting $\partial^k\circ h_j$7 and $\partial^k\circ h_j$8. With the left lexicographic degree-$\partial^k\circ h_j$9 basis

$\mathbb{C}$0

and a diagonal coefficient vector $\mathbb{C}$1, one writes

$\mathbb{C}$2

The first prolongation is multiplication by

$\mathbb{C}$3

and is represented by a matrix $\mathbb{C}$4 through

$\mathbb{C}$5

This realizes the SOS condition for the first prolongation as a linear inequality: $\mathbb{C}$6 Moreover,

$\mathbb{C}$7

The matrix $\mathbb{C}$8 has $\mathbb{C}$9 rows and $(m,f)$0 columns, entries in $(m,f)$1, exactly $(m,f)$2 ones in each column, and between $(m,f)$3 and $(m,f)$4 ones in each row. Its block form is the Macaulay representation stated in the paper: $(m,f)$5 The diagonal identity blocks correspond to the combinatorial splitting

$(m,f)$6

which the paper identifies with the $(m,f)$7-th Macaulay representation of $(m,f)$8.

The underlying commutative-algebra input is Macaulay’s estimate. If $(m,f)$9 is a monomial space in $d$0, and $d$1, then

$d$2

with equality sharp for left lexicographic spaces. In the SOS paper, this estimate is used together with the block structure of $d$3 to convert semipositivity of the prolongation into explicit linear constraints on monomial blocks (Wang et al., 4 Sep 2025).

4. Rank bounds, slack variables, and the SOS conjecture

A key step in the Hermitian SOS application is the grouping

$d$4

which induces a block decomposition

$d$5

With this decomposition, the condition $d$6 is equivalent to

$d$7

The paper introduces counting functions $d$8 for positive, negative, and zero coordinates, with rank $d$9. It then defines slack variables

$J_{n,d}$00

and obtains the rank decompositions

$J_{n,d}$01

and

$J_{n,d}$02

The principal difficulty is the alternating term. To control it, the paper combines Macaulay-type zero-count estimates, the structural sparsity of $J_{n,d}$03, and induction on $J_{n,d}$04.

The resulting theorem is a lower bound for the SOS rank after first prolongation when the original diagonal Hermitian form is not SOS. If $J_{n,d}$05 but $J_{n,d}$06, then for all $J_{n,d}$07,

$J_{n,d}$08

For all $J_{n,d}$09 and $J_{n,d}$10, the stronger estimate

$J_{n,d}$11

is proved. In the notation

$J_{n,d}$12

the same results are stated as $J_{n,d}$13 for all $J_{n,d}$14 and $J_{n,d}$15 for $J_{n,d}$16.

These inequalities are then connected to Ebenfelt’s SOS conjecture and its weak alternative form. In the diagonal case, not necessarily bihomogeneous, the paper decomposes $J_{n,d}$17 into bihomogeneous pieces and uses additivity of ranks: $J_{n,d}$18 This proves the weak alternative SOS conjecture, and hence the SOS conjecture, for $J_{n,d}$19 in the diagonal case. For $J_{n,d}$20, the paper gives partial results, including the universal bound $J_{n,d}$21 and the stronger $J_{n,d}$22 estimate above. The minimal-rank degree-two examples

$J_{n,d}$23

and

$J_{n,d}$24

show that the lower bounds are attained in small dimensions (Wang et al., 4 Sep 2025).

5. Differential-operator Macaulay matrices for GKZ systems and Feynman integrals

In the Feynman-integral application, the Macaulay construction is formulated in the rational Weyl algebra

$J_{n,d}$25

Let $J_{n,d}$26 be a zero-dimensional left ideal generated by Euler and toric operators, or more generally by a holonomic ideal, and assume a finite set of standard monomials $J_{n,d}$27 of size $J_{n,d}$28 is known. Any operator $J_{n,d}$29 has a normal form

$J_{n,d}$30

The Pfaffian relation in the quotient $J_{n,d}$31 for direction $J_{n,d}$32 is written as

$J_{n,d}$33

Expanding the products $J_{n,d}$34 in normal form,

$J_{n,d}$35

produces a Macaulay tensor, which is flattened into a matrix

$J_{n,d}$36

The columns are partitioned into exterior monomials $J_{n,d}$37 and standard monomials $J_{n,d}$38, giving

$J_{n,d}$39

With a corresponding decomposition of $J_{n,d}$40 into coefficient matrices $J_{n,d}$41 and $J_{n,d}$42, the Pfaffian computation becomes the pair of linear systems

$J_{n,d}$43

The algorithm increases a derivative cutoff $J_{n,d}$44, forms

$J_{n,d}$45

constructs the degree-$J_{n,d}$46 Macaulay matrix $J_{n,d}$47, and stops when the exterior system is solvable. Linear algebra over $J_{n,d}$48, often realized through finite-field evaluation and rational reconstruction, then yields the Pfaffian matrices. The paper emphasizes that this avoids non-commutative Gröbner basis computation in $J_{n,d}$49 once $J_{n,d}$50 is known from a commutative Gröbner deformation argument.

This Macaulay matrix then supports two further constructions. First, Pfaffian systems

$J_{n,d}$51

generate contiguity relations and matrix factorial identities used in the holonomic gradient method. Second, the same Pfaffians control the secondary equation for the cohomology intersection matrix,

$J_{n,d}$52

which enables projection of Euler and Feynman integrals onto a chosen basis. The paper reports, for example, an $J_{n,d}$53 case of rank $J_{n,d}$54 with Macaulay matrix size $J_{n,d}$55, where the Pfaffian in direction $J_{n,d}$56 is computed in $J_{n,d}$57 after $J_{n,d}$58 to construct $J_{n,d}$59, and an $J_{n,d}$60 hexagon case of rank $J_{n,d}$61 with a sparse matrix of size about $J_{n,d}$62 and $J_{n,d}$63 to build $J_{n,d}$64 (Chestnov et al., 2022).

6. Computational characteristics, limitations, and methodological relations

The three realizations differ sharply in numerical profile. In the diagonal SOS setting, $J_{n,d}$65 is a sparse $J_{n,d}$66–$J_{n,d}$67 matrix with recursively assembled identity blocks and lower-dimensional prolongation matrices. Its size grows with $J_{n,d}$68 and $J_{n,d}$69, but the analysis depends only on signs and supports through the counts $J_{n,d}$70. The paper therefore characterizes the method as primarily combinatorial and algebraic, and explicitly states that it is robust against numerical conditioning issues because only nonnegativity and support counting are used (Wang et al., 4 Sep 2025).

In the quantum polynomial-solving setting, by contrast, sparse-access oracles and block-encodings for $J_{n,d}$71 and $J_{n,d}$72 are available, and the running time of modern QLS solvers scales as $J_{n,d}$73 in the sparsity $J_{n,d}$74 and condition number $J_{n,d}$75. The obstruction is the lower bound on $J_{n,d}$76, which is exponential in the Hamming weight in the original Macaulay formulation. The paper therefore concludes that in many cases Grover-based exhaustive search outperforms the HHL-based Macaulay approach. It also introduces the Boolean Macaulay matrix by multilinearizing monomials through

$J_{n,d}$77

obtaining a reduced system equivalent to the original one after elimination of the field-equation rows. For this reduced matrix, the lower bound improves to

$J_{n,d}$78

which leaves open a regime of possible superpolynomial speedup when $J_{n,d}$79. The same paper combines Valiant–Vazirani affine hashing with a generalized quantum coupon collector, proving that

$J_{n,d}$80

measurements suffice to recover the support of a unique Boolean solution with probability at least $J_{n,d}$81 (Ding et al., 2021).

Relative to Gröbner-basis and reduction methods, the Macaulay viewpoint is complementary rather than uniform. In the Boolean/cryptanalytic context, classical methods triangulate the Macaulay matrix to compute a Gröbner basis and then solve, whereas the HHL approach directly targets the minimum-$J_{n,d}$82 solution of the Macaulay linear system. In the GKZ/Feynman setting, the Macaulay matrix replaces non-commutative Gröbner basis computations in $J_{n,d}$83 by linear algebra once standard monomials are known, while IBP reductions remain a separate route to linear relations among momentum-space integrals. The Feynman paper also notes trade-offs: degree selection that is too small yields underdetermined systems, degree inflation increases complexity, and singular parameter loci can obstruct recurrence construction or rational reconstruction (Chestnov et al., 2022).

Across these settings, the Macaulay Matrix Method is best understood as a family of basis-dependent linearizations. In one direction it supplies exact combinatorial control of SOS ranks and confirms the diagonal weak alternative form of Ebenfelt’s SOS conjecture for $J_{n,d}$84; in another it exposes a condition-number barrier for quantum linear-system attacks on Boolean polynomial solving; and in another it furnishes an efficient linear-algebraic route to Pfaffian systems, recurrence relations, and intersection-number computations for GKZ and Feynman integrals (Wang et al., 4 Sep 2025, Ding et al., 2021, Chestnov et al., 2022).