Strong Nondegenerate Tensor Tuple

• Strong nondegenerate tensor tuples are collections of tensors that define a polynomial map ensuring component-wise separation and unique solution identification in PCP.
• They provide a sufficient condition for the finiteness property in polynomial complementarity problems by combining an R0-tensor condition with strong nondegeneracy.
• Examples highlight that these tuples can achieve strong nondegeneracy—even with odd-order top tensors—distinguishing them from individual tensor properties.

A strong nondegenerate tensor tuple is a structured tensor tuple $$\Theta=(\mathcal{A}_1,\mathcal{A}_2,\ldots,\mathcal{A}_{m-1})\in \Lambda(m,n)$$ introduced for the study of the polynomial complementarity problem (PCP). Its defining property is that the induced polynomial map $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$ separates points through the component-wise relation

$$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$

The notion generalizes strong nondegeneracy for a single tensor to tuples of tensors of descending orders, and it is used to establish a sufficient condition for the finiteness property of PCP solution sets. In the framework of the polynomial complementarity problem, the principal result is that if $\mathcal{A}_1$ is an $R_0$-tensor and $\Theta$ is a strong non-degenerate tensor tuple, then $\Theta$ has the finiteness property (Sharma et al., 27 Jul 2025).

1. Formal setting and induced polynomial map

The ambient space is $\mathbb{R}^n$, with vectors written in boldface. For a natural number $n$, $[n]=\{1,2,\dots,n\}$. For $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$0, the component-wise product is defined by $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$1 for all $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$2, and for a natural number $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$3, $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$4 is given by $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$5. The set $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$6 denotes real tensors of order $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$7 and dimension $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$8, and

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$9

A tensor tuple of height $$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$0 and dimension $$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$1 is

$$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$2

where $$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$3. The associated polynomial map is

$$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$4

Here $$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$5 has order $$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$6, $$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$7 has order $$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$8, and $$({\bf x}-{\bf y})*\big(\Psi({\bf x})-\Psi({\bf y})\big)={\bf 0}\implies {\bf x}={\bf y}.$$9 is a matrix. For $\mathcal{A}_1$0 and $\mathcal{A}_1$1, the tensor–vector contraction is the homogeneous polynomial vector of degree $\mathcal{A}_1$2 whose $\mathcal{A}_1$3-th component is

$\mathcal{A}_1$4

The polynomial complementarity problem $\mathcal{A}_1$5 is to find $\mathcal{A}_1$6 satisfying

$\mathcal{A}_1$7

Equivalently, with $\mathcal{A}_1$8,

$\mathcal{A}_1$9

The solution set is denoted $R_0$0 (Sharma et al., 27 Jul 2025).

2. Defining strong nondegeneracy

A tensor tuple $R_0$1 is non-degenerate if

$R_0$2

It is strong non-degenerate if

$R_0$3

The latter condition is strictly stronger. A strong non-degenerate tensor tuple is non-degenerate, but the converse does not hold in general. The tuple notion is not obtained by testing the component tensors separately: the paper gives examples in which all component tensors are nondegenerate but the tuple fails to be nondegenerate, and also examples in which the tuple is strong nondegenerate although none of the component tensors is strong nondegenerate. This shows that strong nondegeneracy is intrinsically a property of the polynomial map $R_0$4, not a coordinate-wise aggregation of properties of the individual $R_0$5 (Sharma et al., 27 Jul 2025).

Several structural reductions are immediate. If $R_0$6 for $R_0$7 are zero tensors, then $R_0$8 is non-degenerate if and only if $R_0$9 is a non-degenerate tensor, and $\Theta$0 is strong non-degenerate if and only if $\Theta$1 is a strong non-degenerate tensor. If $\Theta$2 for $\Theta$3 are zero tensors, then $\Theta$4 is non-degenerate if and only if $\Theta$5 is a non-degenerate matrix. In this sense, the tuple framework interpolates between tensor complementarity problems and linear complementarity problems.

A further stability property holds for ordinary nondegeneracy: any principal subtensor tuple of a non-degenerate tensor tuple is non-degenerate. The proof proceeds by embedding a vector on the reduced index set into $\Theta$6 by zero-padding and then invoking the defining implication for $\Theta$7. The paper does not provide an analogous principal-subtensor criterion for strong nondegeneracy (Sharma et al., 27 Jul 2025).

3. Role in the finiteness property of PCP

The central motivation for introducing strong nondegenerate tensor tuples is the finiteness of PCP solution sets. A tensor tuple $\Theta$8 is said to have the finiteness property if $\Theta$9 is a finite set for all $\Theta$0. The main sufficient condition is: $\Theta$1

The $\Theta$2 condition is the tensor analogue of the zero-right-hand-side complementarity condition

$\Theta$3

Its role in the PCP setting is compactness: if $\Theta$4 is an $\Theta$5-tensor, then $\Theta$6 has a compact solution set for any $\Theta$7. The proof of finiteness then combines compactness with strong nondegeneracy. If $\Theta$8 were infinite, one could take an infinite sequence of solutions with a convergent subsequence $\Theta$9. Using the complementarity relations coordinate-wise and continuity of $\mathbb{R}^n$0, one obtains

$\mathbb{R}^n$1

for all sufficiently large $\mathbb{R}^n$2. Strong nondegeneracy then forces $\mathbb{R}^n$3 eventually, so the sequence is eventually constant, contradicting infinitude. This yields finiteness for every $\mathbb{R}^n$4 (Sharma et al., 27 Jul 2025).

The theorem is only sufficient. The paper shows that the converse fails in general: there exist tensor tuples for which $\mathbb{R}^n$5 is $\mathbb{R}^n$6 and $\mathbb{R}^n$7 is finite for all $\mathbb{R}^n$8, yet $\mathbb{R}^n$9 is not strong nondegenerate. It also shows that ordinary nondegeneracy is insufficient for PCP finiteness, even when all component tensors are row-diagonal. These examples delimit the exact scope of the theorem and prevent an identification of nondegeneracy with global finiteness (Sharma et al., 27 Jul 2025).

4. Examples and counterexamples

The examples in the source paper are designed to separate the various notions attached to tensor tuples.

A basic positive example shows that strong nondegeneracy is possible even when the top order $n$0 is odd. Let $n$1, where $n$2 has $n$3 and other entries zero, while $n$4 is the identity. Then for $n$5,

$n$6

so

$n$7

If

$n$8

then $n$9 and

$[n]=\{1,2,\dots,n\}$0

which implies $[n]=\{1,2,\dots,n\}$1. Hence $[n]=\{1,2,\dots,n\}$2 is strong nondegenerate, despite $[n]=\{1,2,\dots,n\}$3 being odd. This is notable because, for single tensors, strong nondegeneracy requires even order, whereas no even-order restriction holds at the tuple level.

A contrasting example exhibits a tensor tuple that is non-degenerate but not strong non-degenerate. For $[n]=\{1,2,\dots,n\}$4, let $[n]=\{1,2,\dots,n\}$5 have

$[n]=\{1,2,\dots,n\}$6

with all other entries zero, and let $[n]=\{1,2,\dots,n\}$7 satisfy

$[n]=\{1,2,\dots,n\}$8

Then

$[n]=\{1,2,\dots,n\}$9

The tuple is non-degenerate, but choosing

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$00

gives

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$01

while $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$02, so the tuple is not strong nondegenerate.

The examples also show that componentwise reasoning is unreliable. One example gives $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$03 with $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$04 and $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$05 individually nondegenerate, but $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$06 fails nondegeneracy because $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$07 satisfies $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$08 with $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$09. Another example gives $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$10 with

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$11

hence

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$12

This tuple is strong nondegenerate, but none of $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$13 is strong nondegenerate. Finally, even if all even-ordered components are strong nondegenerate, $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$14 need not be strong nondegenerate: the paper provides a counterexample in $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$15 with $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$16 and $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$17 strong nondegenerate, yet

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$18

satisfy

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$19

These examples establish that strong nondegeneracy is a genuinely global property of the tuple-induced map $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$20 (Sharma et al., 27 Jul 2025).

5. Relation to TCP, LCP, and adjacent tensor notions

The tuple formalism contains standard complementarity models as special cases. If $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$21 for $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$22 are zero tensors, then $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$23 reduces to the tensor complementarity problem $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$24. If $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$25 for $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$26 are zero tensors, then $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$27 reduces to the linear complementarity problem $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$28. The notion of strong nondegenerate tensor tuple is therefore a PCP-specific extension of strong nondegeneracy from single tensors and matrices.

For TCP, the same paper identifies a distinct equivalence phenomenon on the class of matrix-based tensors. A tensor $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$29, with $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$30 even, is matrix based if there exists a real square matrix $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$31 such that

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$32

In that class, the following are equivalent: $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$33 is non-degenerate; $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$34 is non-degenerate; $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$35 is finite for all $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$36; and $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$37 is finite for all $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$38. This is an equivalence theorem for a special TCP class, not a reformulation of the PCP theorem for tensor tuples.

The broader tensor literature uses related terminology differently. In work on tridimensional tensors, degeneracy is studied via hypermatrices, flattenings, determinantal schemes, kernels of trilinear forms, and hyperdeterminants; that paper states explicitly that it does not introduce “strong nondegeneracy” and instead offers a natural strengthened notion as an interpretation aligned with its framework (Gimigliano et al., 11 May 2026). In work on eigenvectors and singular vector tuples of tensors, “strong nondegeneracy” is again not defined explicitly; the operative notion is Morse-type nondegeneracy of critical points, with a natural strengthened interpretation involving nonsingular Jacobian or bordered KKT matrix and nonzero eigenvalue or singular value (Hu, 2021). These usages are not equivalent to the PCP notion. A common misconception is therefore to treat “strong nondegeneracy” as a uniform tensor concept across subfields. The available sources instead indicate several non-equivalent notions, with the tensor-tuple definition above being specific to complementarity theory.

6. Verification, limitations, and open directions

The paper does not give general algebraic criteria, such as rank-type or determinantal criteria, for checking strong nondegeneracy of tensor tuples. Verification is by direct inspection of the defining implication

$$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$39

for all $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$40, typically by exploiting explicit structure in $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$41. This makes the concept precise but not immediately algorithmic.

No symmetry or positivity assumptions are imposed in the main finiteness theorem; the structural hypothesis is instead the $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$42 property of the leading tensor $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$43, which supplies compactness of the solution set. The proof uses compactness together with an eventually constant sequence argument, rather than polynomial degree theory, Bézout-type bounds, BKK bounds, homotopy, or index arguments. This suggests that the theorem is fundamentally topological-continuity based rather than enumerative.

Several limitations are explicit. Nondegeneracy of a tensor tuple does not imply PCP finiteness, even under row-diagonality. The finiteness theorem does not have a converse in general. Strong nondegeneracy for tuples can occur with odd top order, so any attempt to transplant even-order restrictions from the single-tensor setting to tuples is incorrect. Characterizing strong nondegeneracy by checkable algebraic conditions remains open, as does extending the finiteness theorem without the assumption that $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$44 is $$\Psi({\bf x})=\sum_{k=1}^{m-1}\mathcal{A}_k{\bf x}^{m-k}$$45. A plausible implication is that the main obstacle is not the tuple structure alone, but the need for a robust compactness mechanism compatible with PCP geometry (Sharma et al., 27 Jul 2025).