Polynomial Equivalence in Mathematical Structures

• Polynomial equivalence is a family of notions where objects retain structural properties under polynomial transformations, including shifts, substitutions, and metric maps.
• It is applied in fields like additive combinatorics, algebra, geometry, and coding theory to achieve quantitative bounds and algorithmic reductions.
• Researchers leverage these equivalences to design efficient tests, establish isomorphisms, and analyze both algebraic and combinatorial structures.

Polynomial equivalence is a context-dependent term rather than a single canonical definition. Across the literature it can denote a quantitative equivalence between conjectures with polynomial parameter loss, an algebraic equivalence under variable transformations such as shifts or orthogonal maps, an equivalence of polynomial matrices or quotient rings under structured isomorphisms, a metric equivalence of polynomial functions under bi-Lipschitz maps, or a combinatorial equivalence between polynomial bases expressed through nonnegative expansions (Lovett, 2010, Dvir et al., 2014, Guan et al., 2024, Bodin, 2019, Assaf, 2017).

1. Main senses of the term

The surveyed papers use “polynomial equivalence” in several distinct but related ways. In some settings, “polynomial” refers to the size of parameter losses or algorithmic overhead. In others, it refers to polynomial maps, polynomial substitutions, or polynomially defined objects such as matrices, code families, and generating functions. This suggests that the term is best understood as a family of notions indexed by the structure one wants to preserve.

Context	Objects	Equivalence criterion
Additive combinatorics	conjectures over $\mathbb{F}_2^n$	equivalence with polynomial dependence of parameters (Lovett, 2010)
Variable transformations	$f,g \in \mathbb{F}[x_1,\dots,x_n]$	$f(x+a)\equiv g(x)$ or $f(Rx)=g(x)$ (Dvir et al., 2014, Helmer et al., 5 Jan 2026)
Matrix and ring theory	polynomial matrices, code rings, matrices	unimodular equivalence, $x\mapsto \alpha x$, or $A=f(B), B=g(A)$ (Guan et al., 2024, Ou-azzou et al., 6 Mar 2025, Zhang et al., 4 Oct 2025)
Metric geometry and singularities	polynomial maps and germs	bi-Lipschitz or semialgebraic bi-Lipschitz equivalence (Bodin, 2019, Alvarez, 8 Mar 2025)
Dynamics and combinatorics	orbital equations, polynomial bases	Tschirnhausen transformations or positivity-preserving basis translation (Brison et al., 2018, Assaf, 2017)

A recurring pattern is that one asks whether two polynomially described objects can be interconverted without leaving a prescribed structural class. The allowed conversions vary sharply from one area to another, and so do the invariants that control equivalence.

2. Quantitative polynomial equivalence in additive combinatorics

In additive combinatorics, the phrase refers to quantitative equivalence of conjectures with polynomial parameter loss. Lovett studies two finite-field conjectures over $\mathbb{F}_2^n$: the Polynomial Freiman–Ruzsa conjecture, which concerns sets $S$ with small doubling $|S+S|\le K|S|$, and the Polynomial inverse Gowers $U^3$ conjecture, which concerns Boolean functions $f,g \in \mathbb{F}[x_1,\dots,x_n]$0 with $f,g \in \mathbb{F}[x_1,\dots,x_n]$1. The first predicts a subset $f,g \in \mathbb{F}[x_1,\dots,x_n]$2 of size at least $f,g \in \mathbb{F}[x_1,\dots,x_n]$3 whose span has size at most $f,g \in \mathbb{F}[x_1,\dots,x_n]$4; the second predicts a quadratic polynomial $f,g \in \mathbb{F}[x_1,\dots,x_n]$5 with $f,g \in \mathbb{F}[x_1,\dots,x_n]$6 (Lovett, 2010).

The main theorem is that these two conjectures are equivalent in a strong parameter sense. Assuming Polynomial Freiman–Ruzsa, the only super-polynomial loss in Samorodnitsky’s inverse $f,g \in \mathbb{F}[x_1,\dots,x_n]$7 proof disappears, so the final correlation advantage becomes polynomial in $f,g \in \mathbb{F}[x_1,\dots,x_n]$8. In the reverse direction, Lovett constructs from an approximate homomorphism $f,g \in \mathbb{F}[x_1,\dots,x_n]$9 a Boolean function $f(x+a)\equiv g(x)$0 on $f(x+a)\equiv g(x)$1, proves $f(x+a)\equiv g(x)$2, applies the inverse $f(x+a)\equiv g(x)$3 conjecture to obtain a quadratic approximation, extracts a linear map $f(x+a)\equiv g(x)$4, and then upgrades frequent approximate additivity to a global description $f(x+a)\equiv g(x)$5 with $f(x+a)\equiv g(x)$6 (Lovett, 2010).

Here “equivalence” is not mere logical equivalence. It is quantitative: polynomial bounds on one side induce polynomial bounds on the other side, uniformly in dimension. The result identifies the difficulty of replacing exponential losses by polynomial losses in two ostensibly different problems—small-doubling structure and inverse Gowers theory—as essentially the same problem.

3. Equivalence under changes of variables

A classical algebraic-complexity meaning of polynomial equivalence is equivalence under shifts of variables. For $f(x+a)\equiv g(x)$7, they are shift-equivalent if there exists $f(x+a)\equiv g(x)$8 such that $f(x+a)\equiv g(x)$9. The corresponding Shift Equivalence Testing problem asks for a decision procedure and, when possible, an explicit shift vector. Over characteristic $f(Rx)=g(x)$0 or characteristic $f(Rx)=g(x)$1, Dvir, Oliveira, and Shpilka show that shift-equivalence testing is reducible to Polynomial Identity Testing under mild closure assumptions on the circuit classes involved; in particular, there is a randomized black-box algorithm with running time $f(Rx)=g(x)$2, and derandomizing shift-equivalence testing is essentially equivalent to derandomizing PIT in both black-box and white-box settings (Dvir et al., 2014).

The algorithmic mechanism is multivariate Taylor expansion together with directional derivatives. Writing $f(Rx)=g(x)$3 into homogeneous parts, one obtains linear conditions on the unknown shift vector by comparing the degree-$f(Rx)=g(x)$4, degree-$f(Rx)=g(x)$5, and then lower-degree components of $f(Rx)=g(x)$6 and $f(Rx)=g(x)$7. A crucial “domino effect” says that equality of first directional derivatives forces equality of all higher directional derivatives. This turns apparently nonlinear conditions into a sequence of linear dependence problems among implicitly given polynomials, which PIT can solve (Dvir et al., 2014).

A nearby but much harder problem is whether a single polynomial can be made sparser by a shift. For a ring $f(Rx)=g(x)$8, $f(Rx)=g(x)$9 asks whether there exists $x\mapsto \alpha x$0 such that $x\mapsto \alpha x$1 has fewer monomials than $x\mapsto \alpha x$2. Fomin, Golovnev, and Kulikov reduce Hilbert’s Nullstellensatz over $x\mapsto \alpha x$3 to this sparsification-by-shift problem, proving that $x\mapsto \alpha x$4 is undecidable, that $x\mapsto \alpha x$5 is $x\mapsto \alpha x$6-complete in the Blum–Shub–Smale model whenever $x\mapsto \alpha x$7 is $x\mapsto \alpha x$8-complete for an integral domain $x\mapsto \alpha x$9 that is not a field, and that several gap versions are also hard, including $A=f(B), B=g(A)$0-gap hardness over $A=f(B), B=g(A)$1, and $A=f(B), B=g(A)$2 for every $A=f(B), B=g(A)$3 (Chillara et al., 2022).

Taken together, these results separate two nearby-looking notions. Testing whether two explicitly given polynomials differ by a shift can be PIT-like, while asking whether some shift exposes a sparse representation can be as hard as polynomial-system satisfiability.

4. Algebraic equivalence of polynomial matrices, code families, and matrices

For multivariate polynomial matrices over $A=f(B), B=g(A)$4, the standard notion is left-right unimodular equivalence: $A=f(B), B=g(A)$5 if $A=f(B), B=g(A)$6 for $A=f(B), B=g(A)$7 invertible over $A=f(B), B=g(A)$8. Li studies square matrices with $A=f(B), B=g(A)$9, where $\mathbb{F}_2^n$0 is irreducible, proves a generalized global-local theorem of Vaserstein type, and shows that such matrices are equivalent to their Smith forms exactly when all reduced-minor ideals satisfy $\mathbb{F}_2^n$1. In particular, for $\mathbb{F}_2^n$2 matrices with $\mathbb{F}_2^n$3, the condition $\mathbb{F}_2^n$4 implies $\mathbb{F}_2^n$5 (Guan et al., 2024).

A stricter matrix-theoretic variant is semi-scalar equivalence. Two nonsingular polynomial matrices $\mathbb{F}_2^n$6 are semi-scalar equivalent if $\mathbb{F}_2^n$7 for a constant $\mathbb{F}_2^n$8 and a polynomially invertible $\mathbb{F}_2^n$9. Kazimirskiĭ and Petrychkovych give necessary and sufficient conditions in terms of a homogeneous linear system $S$0, where $S$1 is the last invariant factor in the Smith form and the existence of a solution yielding a nonsingular reshaped matrix $S$2 is equivalent to semi-scalar equivalence. They also show that similarity of families $S$3 and $S$4 over $S$5 is equivalent to semi-scalar equivalence of the associated monic matrix polynomials $S$6 and $S$7 (Prokip, 2020).

A different use of the term appears for ordinary matrices over a field of characteristic $S$8. Two matrices $S$9 are polynomial equivalent if there exist polynomials $|S+S|\le K|S|$0 such that $|S+S|\le K|S|$1 and $|S+S|\le K|S|$2. Wang proves that $|S+S|\le K|S|$3 if and only if $|S+S|\le K|S|$4 and $|S+S|\le K|S|$5 are polynomial equivalent. For balanced matrices, including nilpotent matrices, equality of clifforders $|S+S|\le K|S|$6 is equivalent to odd polynomial equivalence, and in the quasi-commutative setting $|S+S|\le K|S|$7 this is extended to $|S+S|\le K|S|$8-polynomial equivalence and $|S+S|\le K|S|$9-equivalence (Zhang et al., 4 Oct 2025).

Coding theory contributes another structurally rigid notion. For polycyclic codes over $U^3$0, equivalence of families is defined by an isometric $U^3$1-algebra isomorphism between ambient quotient rings induced by $U^3$2. This yields $U^3$3-equivalence for $U^3$4-trinomial families and a general $U^3$5-equivalence for arbitrary polycyclic families. The paper computes the number of equivalence classes and proves, for example, that when $U^3$6, any $U^3$7-trinomial code family is equivalent to the one defined by $U^3$8 (Ou-azzou et al., 6 Mar 2025).

5. Geometric and metric equivalence of polynomial functions

In Lipschitz geometry, equivalence means preservation of metric structure rather than algebraic form. Fernandes and Ruas study the family $U^3$9 over $f,g \in \mathbb{F}[x_1,\dots,x_n]$00 or $f,g \in \mathbb{F}[x_1,\dots,x_n]$01. They show that if $f,g \in \mathbb{F}[x_1,\dots,x_n]$02, then $f,g \in \mathbb{F}[x_1,\dots,x_n]$03 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$04 are not right-bilipschitz equivalent at infinity in the real case, and for the modified family $f,g \in \mathbb{F}[x_1,\dots,x_n]$05 with $f,g \in \mathbb{F}[x_1,\dots,x_n]$06, they are not left-right bilipschitz equivalent either. At the same time, individual fibers can still be bilipschitz equivalent: the level sets $f,g \in \mathbb{F}[x_1,\dots,x_n]$07 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$08, as well as $f,g \in \mathbb{F}[x_1,\dots,x_n]$09 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$10, are bilipschitz equivalent as subsets of $f,g \in \mathbb{F}[x_1,\dots,x_n]$11. In the complex case, the same family can be topologically equivalent while failing bilipschitz equivalence, showing that bilipschitz classification is strictly finer than topological classification (Bodin, 2019).

A semialgebraic refinement is developed for quasihomogeneous real polynomial germs $f,g \in \mathbb{F}[x_1,\dots,x_n]$12. Fix $f,g \in \mathbb{F}[x_1,\dots,x_n]$13; if $f,g \in \mathbb{F}[x_1,\dots,x_n]$14 is $f,g \in \mathbb{F}[x_1,\dots,x_n]$15-quasihomogeneous of degree $f,g \in \mathbb{F}[x_1,\dots,x_n]$16, then the associated height functions are $f,g \in \mathbb{F}[x_1,\dots,x_n]$17 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$18. Alvarez classifies one-variable polynomials up to Lipschitz equivalence by degree, critical-point multiplicities, and multiplicity symbols, then proves that under broad hypotheses semialgebraic bi-Lipschitz equivalence of $f,g \in \mathbb{F}[x_1,\dots,x_n]$19 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$20 implies Lipschitz equivalence of their height functions. The key tools are $f,g \in \mathbb{F}[x_1,\dots,x_n]$21-transforms and inverse $f,g \in \mathbb{F}[x_1,\dots,x_n]$22-transforms, which convert a semialgebraic bi-Lipschitz germ into compatible one-variable bi-Lipschitz data and back again. Applied to the Henry–Parusiński family $f,g \in \mathbb{F}[x_1,\dots,x_n]$23, the method shows that semialgebraic bi-Lipschitz equivalence of real quasihomogeneous polynomials also admits continuous moduli (Alvarez, 8 Mar 2025).

Orthogonal equivalence gives yet another geometric notion. For $f,g \in \mathbb{F}[x_1,\dots,x_n]$24, one asks for $f,g \in \mathbb{F}[x_1,\dots,x_n]$25 such that $f,g \in \mathbb{F}[x_1,\dots,x_n]$26. Berrone, Gaitsgory, and Yekhanin introduce Polynomial-Weighted Principal Component Analysis, defined from the covariance matrix of the homogenized polynomial weighted by $f,g \in \mathbb{F}[x_1,\dots,x_n]$27 on the sphere. Under a generic distinct-eigenvalue assumption, PW-PCA reduces orthogonal equivalence to a finite sign-flip search: if $f,g \in \mathbb{F}[x_1,\dots,x_n]$28 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$29 are the PW-PCA decompositions, then $f,g \in \mathbb{F}[x_1,\dots,x_n]$30 if and only if there exists $f,g \in \mathbb{F}[x_1,\dots,x_n]$31 such that $f,g \in \mathbb{F}[x_1,\dots,x_n]$32, and then $f,g \in \mathbb{F}[x_1,\dots,x_n]$33 is a certificate of orthogonal equivalence (Helmer et al., 5 Jan 2026).

6. Dynamical, combinatorial, and complexity-geometric uses

In one dynamical usage, polynomial equivalence concerns orbital equations generated by iteration of polynomial maps. Gallas studies cyclic quintics $f,g \in \mathbb{F}[x_1,\dots,x_n]$34 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$35, both with discriminant $f,g \in \mathbb{F}[x_1,\dots,x_n]$36, and shows that their orbital equations are not uniquely represented. The paper gives five direct and five inverse polynomial transformations between the roots of $f,g \in \mathbb{F}[x_1,\dots,x_n]$37 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$38, and recursive chains of transformations produce infinitely many equivalent orbital equations, all connected by nonlinear transformations that preserve the field structure and orbit data (Gallas, 2018).

A complementary detector for the same phenomenon is standard polynomial interpolation. For irreducible $f,g \in \mathbb{F}[x_1,\dots,x_n]$39 of the same degree, one numerically orders the roots of $f,g \in \mathbb{F}[x_1,\dots,x_n]$40 and $f,g \in \mathbb{F}[x_1,\dots,x_n]$41, interpolates the polynomial $f,g \in \mathbb{F}[x_1,\dots,x_n]$42 sending the chosen ordering of roots of $f,g \in \mathbb{F}[x_1,\dots,x_n]$43 to a permutation of the roots of $f,g \in \mathbb{F}[x_1,\dots,x_n]$44, and then searches for permutations producing rational or integral coefficients. This yields explicit Tschirnhausen transformations and hence explicit field isomorphisms. In the Vandermonde quintic example, interpolation recovered the full set of ten transformations faster and with less memory than systematic search; the paper also resolves Hasse’s quintic problem symmetrically and maps equivalence classes of several sextic families (Brison et al., 2018).

In algebraic combinatorics, the phrase appears in a weaker but structurally important sense. Assaf’s weak dual equivalence does not assert equality of polynomial families; instead it produces nonnegative basis translations. Dual equivalence gives a short combinatorial proof that Stanley symmetric functions are Schur positive, while weak dual equivalence gives a short combinatorial proof that Schubert polynomials are key positive. The same framework yields a nonnegative Littlewood–Richardson rule for the product of a key polynomial and a Schur polynomial, and skew key polynomials that, when skewed by a partition, expand nonnegatively in the key basis (Assaf, 2017).

A final quantitative usage appears in quantum complexity geometry. Brown studies right-invariant metrics on the unitary group and proves the polynomial equivalence of a broad class of “complexity geometries.” Within the resulting universality classes, any unitary reachable in one metric can be approximated in any other metric in the class with slowdown at worst polynomial in the path length and number of qubits and inverse-polynomial in the permitted error. The paper distinguishes Killing-distance error and operator-norm error, proves exponential diameter for all metrics in both equivalence classes, and shows that metrics in the operator-norm class also give an alternative definition of BQP (Brown, 2022).

Across these examples, “polynomial equivalence” ranges from exact algebraic convertibility to metric, combinatorial, or quantitative simulation. The common thread is preservation of a chosen structure under a restricted class of polynomially controlled transformations; the object preserved may be a conjectural parameter regime, a circuit or ring representation, a metric germ, an orbit, a basis expansion, or a complexity geometry.