Nondegenerate Tensor Tuple

• Nondegenerate tensor tuple is a framework that imposes joint regularity on multiple tensors by excluding hidden directions, ensuring uniqueness and finiteness in solutions across diverse contexts.
• It generalizes classical nondegeneracy by treating the collection as a single structured object, linking polynomial complementarity, spectral theory, algebraic geometry, and differential geometry.
• This concept underpins robust tensor reconstruction, stability in eigenvector analysis, and controlled symmetry in geometric structures, offering practical insights into tensor behavior.

“Nondegenerate tensor tuple” is not a single universal notion but a family of context-dependent nondegeneracy conditions for collections of tensors, tensor-derived equations, or tensor-critical configurations. In the polynomial complementarity literature, a tensor tuple is $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in \Lambda(m,n)$$ with associated polynomial map $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$, and nondegeneracy means ${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$, while strong nondegeneracy means $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$ (Sharma et al., 27 Jul 2025). In intrinsic tensor spectral theory, the same expression is naturally attached to eigenvectors or singular tuples regarded as nondegenerate critical points of the tensor’s canonical function on a sphere or product of spheres (Basso et al., 2020, Hu, 2021). In algebraic-geometric and differential-geometric settings, related usages refer to finite reduced singular-tuple configurations, or to joint tensor configurations such as $(J,N_J,h,\zeta)$ or $(H,E_a,L_a)$ whose stabilizers and deformation theory control the geometry (Turatti, 2021, Kruglikov et al., 2015, Gregorovič et al., 2024).

1. Complementarity-theoretic definition

The most explicit stand-alone definition appears in the theory of the polynomial complementarity problem. For $m,n\in\mathbb N$, one sets

$$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$

and for $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$ defines

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$

The tuple $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$0 is called a nondegenerate tensor tuple if

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$1

and a strong nondegenerate tensor tuple if

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$2

Every strong nondegenerate tensor tuple is nondegenerate, and these notions generalize both nondegenerate tensors in the tensor complementarity problem and nondegenerate matrices in the linear complementarity problem (Sharma et al., 27 Jul 2025).

The reduction to earlier notions is exact. If $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$3, then $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$4, so tuple nondegeneracy is equivalent to the classical condition for a single tensor. If $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$5, then $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$6, and tuple nondegeneracy is equivalent to $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$7 being a nondegenerate matrix, i.e. all principal minors are nonzero. This places the tuple notion on the same axis as the classical LCP–TCP–PCP progression.

A central point is that nondegeneracy is a property of the tuple as a whole, not of its components separately. The cited examples show that all component tensors may be nondegenerate while the tuple is not; conversely, a tuple may be nondegenerate even though one component tensor is degenerate. The strong notion is equally non-componentwise: a strong nondegenerate tuple need not have strong nondegenerate components, and even if all even-ordered component tensors are strong nondegenerate, the tuple may fail to be strong nondegenerate. The theory therefore treats $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$8 as an indivisible structured object rather than as a coordinatewise checklist.

This complementarity notion is tied directly to finiteness of solution sets. If $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$9 is an ${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$0-tensor and ${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$1 is a strong nondegenerate tensor tuple, then ${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$2 has the finiteness property: ${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$3 The proof combines compactness from the ${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$4-condition with an accumulation-point argument that forces eventual equality of distinct solutions under strong nondegeneracy. The converse fails: finiteness for all right-hand sides does not imply strong nondegeneracy, and ordinary tuple nondegeneracy is not equivalent to finiteness even in row-diagonal PCPs. The paper also proves that principal subtensor tuples of a nondegenerate tensor tuple are again nondegenerate, mirroring the matrix fact that principal submatrices of a nondegenerate matrix are nondegenerate (Sharma et al., 27 Jul 2025).

2. Singular tuples as nondegenerate critical points

In intrinsic tensor analysis, a tensor is identified with its associated multilinear map rather than with a coordinate array. For

${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$5

the associated map is

${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$6

On the product of unit spheres

${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$7

the singular vectors of ${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$8 are the critical points of ${\bf x}*\Psi({\bf x})=0\Rightarrow {\bf x}=0$9, equivalently the tuples $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$0 satisfying

$$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$1

For a symmetric tensor $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$2, the corresponding unit eigenvectors are the critical points of $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$3, and $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$4 is called nondegenerate if all its eigenvectors are nondegenerate critical points of that restricted function (Basso et al., 2020).

In this framework, nondegeneracy is Morse nondegeneracy. A critical point is nondegenerate when the Hessian of the restricted function on the tangent space is nonsingular. The paper explicitly formulates this for symmetric eigenvectors, and the same source states that the natural extension to singular tuples on products of spheres is to require the Hessian of $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$5 at the tuple to be nonsingular. This reformulation makes nondegeneracy coordinate-free, invariant under orthonormal basis changes, and directly accessible to Morse theory. For symmetric tensors, the index of a nondegenerate eigenvector is the Morse index of the corresponding critical point, and the Morse inequalities on $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$6 translate into constraints on the numbers of eigenvectors of each index. In particular, if $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$7 is symmetric and nondegenerate, then it has at least one eigenvector of index $$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$8 (Basso et al., 2020).

A second paper makes this critical-point notion explicit for both singular vector tuples and Z-eigenvectors. For

$$({\bf x}-{\bf y})*(\Psi({\bf x})-\Psi({\bf y}))=0\Rightarrow {\bf x}={\bf y}$$9

on

$(J,N_J,h,\zeta)$0

a singular vector tuple is nondegenerate if the Riemannian Hessian $(J,N_J,h,\zeta)$1 is nonsingular. For a symmetric tensor, a Z-eigenvector is nondegenerate if it is a nondegenerate critical point of

$(J,N_J,h,\zeta)$2

on the sphere; equivalently, the Jacobian of

$(J,N_J,h,\zeta)$3

is nonsingular at $(J,N_J,h,\zeta)$4. The main generic statement is that each singular vector tuple and each Z-eigenvector of a generic tensor is nondegenerate. The same paper also proves that each nonzero singular vector tuple of an orthogonally decomposable tensor, and each nonzero Z-eigenvector of a symmetric orthogonally decomposable tensor, is nondegenerate (Hu, 2021).

3. Algebraic-geometric singular-tuple loci

Over $(J,N_J,h,\zeta)$5, the singular tuples of a multisymmetric tensor are encoded as a projective zero locus. For

$(J,N_J,h,\zeta)$6

the singular tuple equations are

$(J,N_J,h,\zeta)$7

and the locus of projective solutions is a zero-dimensional subscheme

$(J,N_J,h,\zeta)$8

The paper realizes $(J,N_J,h,\zeta)$9 as the zero locus $(H,E_a,L_a)$0 of a section of a vector bundle $(H,E_a,L_a)$1, and for a general tensor states that this scheme is finite and reduced, with length equal to the ED-degree of the Segre–Veronese variety. In that setting, the paper does not formally define “nondegenerate singular tuple,” but it treats the generic finite reduced configuration as the relevant nondegenerate regime (Turatti, 2021).

The main theorem is a reconstruction result from singular tuples. Assuming the triangular inequality

$(H,E_a,L_a)$2

for every $(H,E_a,L_a)$3 with $(H,E_a,L_a)$4, and excluding $(H,E_a,L_a)$5 in the $(H,E_a,L_a)$6 case, a general tensor is projectively unique from its singular tuple locus if at least one degree $(H,E_a,L_a)$7 is odd: $(H,E_a,L_a)$8 If all degrees are even, the fiber is a projective line,

$(H,E_a,L_a)$9

so the only ambiguity is the quadratic product direction. The parity split is governed by the kernel of the map $m,n\in\mathbb N$0 from tensors to sections: $m,n\in\mathbb N$1 is injective if some $m,n\in\mathbb N$2 is odd, while

$m,n\in\mathbb N$3

when all $m,n\in\mathbb N$4 are even (Turatti, 2021).

A related order-$m,n\in\mathbb N$5 projective theory studies the set

$m,n\in\mathbb N$6

For a generic tensor, the number of complex singular $m,n\in\mathbb N$7-tuples is finite and equals the ED-degree $m,n\in\mathbb N$8, depending only on the format. The same source states that these points are simple and therefore isolated nondegenerate critical points. It then analyzes the projective span $m,n\in\mathbb N$9 and the linear critical space $$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$0, proving that $$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$1 in sub-boundary formats and exhibiting stabilization phenomena for the span dimension beyond boundary format. In the special families $$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$2, $$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$3, and $$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$4 with $$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$5, it proves that a generic tensor belongs to the span of its singular tuples,

$$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$6

and conjectures that this holds for every tensor format. This suggests a broader algebraic meaning of nondegeneracy: the singular-tuple configuration is finite, simple, and linearly rich enough to recover the tensor generically (Sodomaco et al., 2022).

4. Joint tensor tuples in almost complex six-manifolds

In differential geometry, the phrase refers not to complementarity or Morse criticality but to a joint tensor configuration whose stabilizer controls symmetry. For a $$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$7-dimensional almost complex manifold $$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$8, the Nijenhuis tensor

$$\Lambda(m,n):=\mathbb T(m,n)\times\mathbb T(m-1,n)\times\cdots\times \mathbb T(2,n),$$9

is called nondegenerate when it is a $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$0-antilinear isomorphism of real vector spaces. At each point, one may regard the tangent space as a complex $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$1-dimensional vector space $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$2 with a nondegenerate antilinear skew-symmetric map

$$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$3

From $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$4, the paper canonically constructs a Hermitian form

$$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$5

and a complex $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$6-form

$$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$7

When both are nondegenerate, the symmetry of $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$8 preserves the resulting metric and holomorphic volume form, so the stabilizer is contained in $$\Theta=(\mathcal A_1,\ldots,\mathcal A_{m-1})\in\Lambda(m,n)$$9 or $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$0 (Kruglikov et al., 2015).

In this setting, the relevant tensor tuple is

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$1

Its nondegeneracy means that $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$2 is an isomorphism, $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$3 is a nondegenerate Hermitian form of signature $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$4 or $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$5, and $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$6 is a nonvanishing complex $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$7-form. The classification of automorphism groups is then driven by this pointwise algebraic structure. Earlier work cited in the paper gives the absolute bound

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$8

with equality only in the $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}.$$9-symmetric cases. The paper itself proves the sub-maximal statement: if $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$00 is not locally $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$01-symmetric, then

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$02

and equality occurs precisely for the homogeneous spaces

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$03

all strictly nearly (pseudo-)Kähler. It also proves that nondegenerate almost complex structures with $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$04-dimensional symmetry are locally homogeneous with semisimple stabilizer $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$05 or $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$06 (Kruglikov et al., 2015).

The role of nondegeneracy here is therefore representation-theoretic. The tensor tuple does not encode a complementarity map or a critical-point Hessian; it encodes a rigid $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$07-structure at a point, and nondegeneracy is the condition that the joint stabilizer of the tuple is sharply constrained.

5. Levi-form-based tensor tuples in $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$08-nondegenerate CR geometry

For uniformly $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$09-nondegenerate CR hypersurfaces, the relevant tensor tuple is built from the Levi form, the Levi kernel, and the modified symbol. At a point $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$10, one has the Levi quotient $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$11 and the Levi kernel $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$12. In adapted frames, the Levi form is represented by a nondegenerate Hermitian matrix $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$13, while each kernel vector $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$14 determines a symmetric matrix

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$15

The paper proves that a CR hypersurface of dimension $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$16 with Levi rank $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$17 is uniformly $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$18-nondegenerate if and only if $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$19 and the matrices

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$20

are linearly independent. Equivalently, the map

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$21

is injective and has image of dimension $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$22 (Gregorovič et al., 2024).

The canonical model theory packages this into a pair $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$23. Every $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$24-nondegenerate model is locally equivalent to one of the form

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$25

with $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$26 a constant nondegenerate Hermitian $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$27 matrix and $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$28 a holomorphic symmetric matrix-valued function vanishing at $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$29, such that

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$30

are linearly independent in $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$31. Two such models are equivalent if and only if there exist $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$32 and a biholomorphism $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$33 fixing $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$34 such that

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$35

This makes $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$36 the model-level tensor tuple, while the first derivatives of $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$37 reproduce the $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$38 (Gregorovič et al., 2024).

The modified symbol adds a further family of degree-zero tensors $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$39, so that the full pointwise invariant can be viewed as

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$40

The paper characterizes abstract modified symbols algebraically and proves that every point in a uniformly $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$41-nondegenerate CR hypersurface is canonically associated with such a model structure. It also shows that for each $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$42 the moduli space of $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$43-nondegenerate CR hypersurface models in $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$44 is infinite dimensional, and that these models automatically possess infinitesimal symmetries spanning a complement to their Levi kernel (Gregorovič et al., 2024).

Here again, nondegeneracy is a joint condition on a tensor package. The Levi form alone is insufficient; the higher tensor $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$45, and for modified symbols the $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$46, are the additional data that replace the single nondegenerate Levi form of the Levi-nondegenerate case.

6. Comparative interpretation

The literature therefore uses “nondegenerate tensor tuple” in several non-equivalent senses. The common object may be a polynomial map, a critical configuration, a projective singular-tuple scheme, or a pointwise $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$47-structure.

Context	Tensor tuple/object	Nondegeneracy condition
PCP/TCP/LCP	$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$48	$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$49; strong version uses pairs $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$50
Spectral/Morse theory	singular tuple or eigenpair	Hessian of the restricted function on the sphere/product of spheres is nonsingular
Multisymmetric/ED geometry	$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$51 or $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$52	finite reduced singular-tuple configuration; simple critical points; sometimes reconstruction of $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$53
Almost complex $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$54-geometry	$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$55	$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$56 is an isomorphism, $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$57 nondegenerate, $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$58 nonvanishing
$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$59-nondegenerate CR geometry	$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$60 or $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$61	$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$62 nondegenerate and the $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$63 are linearly independent

Several misconceptions are ruled out by the sources. First, nondegeneracy is not generally a componentwise property: in the PCP setting, the tuple can fail to be nondegenerate although each component tensor is nondegenerate, and a strong nondegenerate tuple need not have strong nondegenerate components (Sharma et al., 27 Jul 2025). Second, nondegeneracy is not uniformly equivalent to finiteness: for PCP, ordinary nondegeneracy is not equivalent to finiteness of all solution sets, whereas strong nondegeneracy together with an $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$64-condition is sufficient but not necessary (Sharma et al., 27 Jul 2025). Third, in algebraic reconstruction problems, parity can force a controlled one-dimensional ambiguity even in otherwise generic situations, through the quadratic product direction in the all-even multisymmetric case (Turatti, 2021).

A plausible unifying interpretation is that nondegeneracy excludes hidden directions. In complementarity theory, it excludes nontrivial vectors complementary to $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal A_k{\bf x}^{m-k}$$65. In Morse-theoretic tensor spectral theory, it excludes Hessian null directions at a singular tuple. In algebraic-geometric reconstruction, it excludes multiplicities or invisible kernel directions in the map from tensors to singular-tuple loci. In differential geometry, it excludes stabilizer enlargement by forcing a rigid joint tensor package. The term is therefore best understood as a family resemblance concept: the precise definition depends on the ambient problem, but in each case it identifies a tensor tuple whose associated equations, symmetries, or critical configurations are maximally regular.