Sign Preservers: Theory and Applications

• Sign Preservers are mathematical maps or operators that maintain the intrinsic sign structure of functions, matrices, and operators, ensuring invariance in properties like determinants and minor signs.
• They are systematically classified across several contexts, including Schur matrix functionals, sign regularity in matrices, and positivity-preserving transforms in elliptic and differential operators.
• Applications span polynomial root analysis, stability in numerical schemes, and theoretical insights into matrix theory, impacting both abstract research and practical modeling.

In the literature surveyed here, closely related terminology is used for several distinct preservation problems centered on sign-structured data. These include linear maps preserving a Schur matrix functional such as the determinant or permanent, operators preserving sign regularity of minors, entrywise transforms preserving positive definiteness or positive semidefiniteness, fourth-order elliptic operators sending non-positive data to negative solutions, and reconstructions whose interface jumps preserve the sign of cell jumps in entropy-stable schemes (Pazzis, 2018, Choudhury et al., 2024, Guillot et al., 8 Jul 2025, Laurencot et al., 2013, Fjordholm et al., 2015). A persistent source of ambiguity is that, in the Descartes-rule literature, a “sign preservation” is not a preserver map at all, but an adjacent equality in a coefficient sign pattern (Cheriha et al., 2019).

1. Terminological scope and Descartes-type sign preservations

For a monic real polynomial

$P(x)=x^d+\sum_{j=0}^{d-1}a_jx^j,\qquad a_j\neq 0,$

the sign pattern is

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$

A sign change is an index $j$ with $\sigma_{j-1}\neq \sigma_j$, while a sign preservation is an index $j$ with $\sigma_{j-1}=\sigma_j$. If $c$ and $p$ denote the numbers of sign changes and sign preservations, then $c+p=d$. Descartes’ rule yields

$pos\le c,\qquad c-pos\equiv 0 \pmod 2,$

and, after applying the rule to $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$0,

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$1

Thus sign preservations are the Descartes-dual quantity controlling negative roots (Cheriha et al., 2019).

For hyperbolic polynomials with all coefficients nonzero, the count becomes exact: the number of positive roots equals $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$2, and the number of negative roots equals $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$3. Under the additional assumption that all root moduli are distinct, the problem becomes finer: one asks which words in $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$4 can occur as the order of moduli of positive and negative roots. The change–preservation pattern, a word in $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$5, and the order of moduli form a compatible couple when the number of $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$6’s matches the number of $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$7’s and the number of $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$8’s matches the number of $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$9’s. Every sign pattern is realizable with its canonical order, obtained by reading the change–preservation pattern from the right and replacing $j$0, $j$1. Canonical sign patterns are exactly those whose change–preservation word contains no subword $j$2 or $j$3 (Kostov, 2023).

The same theme appears in refined Descartes problems for patterns with exactly two sign changes. Writing

$j$4

one has $j$5 and $j$6. Here the total number $j$7 is not decisive by itself: the block structure of sign preservations can force or forbid realizability of extremal pairs $j$8. In particular, the sufficient condition

$j$9

guarantees realizability with $\sigma_{j-1}\neq \sigma_j$0, whereas patterns $\sigma_{j-1}\neq \sigma_j$1 with $\sigma_{j-1}\neq \sigma_j$2 and $\sigma_{j-1}\neq \sigma_j$3 are not realizable with $\sigma_{j-1}\neq \sigma_j$4 (Cheriha et al., 2019). This suggests that, even in the non-operator sense, “sign preservation” is already a structural constraint rather than a mere count.

2. Schur matrix functionals, determinant preservers, and permanent preservers

A systematic linear-preserver formulation appears for Schur matrix functionals. Given a field $\sigma_{j-1}\neq \sigma_j$5, $\sigma_{j-1}\neq \sigma_j$6, and a nonvanishing map

$\sigma_{j-1}\neq \sigma_j$7

the associated Schur matrix functional is

$\sigma_{j-1}\neq \sigma_j$8

The determinant corresponds to $\sigma_{j-1}\neq \sigma_j$9, and the permanent to the constant map $j$0. For two such maps $j$1, an $j$2-transformation is a linear map $j$3 such that

$j$4

A key structural fact is that every $j$5-transformation is automatically an automorphism of $j$6, and its inverse is a $j$7-transformation (Pazzis, 2018).

In the determinant case, the classification is exactly Frobenius–Dieudonné. The linear maps satisfying

$j$8

are precisely

$j$9

with $\sigma_{j-1}=\sigma_j$0 and $\sigma_{j-1}=\sigma_j$1. If one allows a scalar factor $\sigma_{j-1}=\sigma_j$2,

$\sigma_{j-1}=\sigma_j$3

the same forms occur with $\sigma_{j-1}=\sigma_j$4. Over $\sigma_{j-1}=\sigma_j$5, exact determinant preservation also preserves the sign of the determinant (Pazzis, 2018).

The general theory organizes existence of $\sigma_{j-1}=\sigma_j$6-transformations by PH-equivalence: such a transformation exists exactly when $\sigma_{j-1}=\sigma_j$7 is PH-equivalent to $\sigma_{j-1}=\sigma_j$8 or to $\sigma_{j-1}=\sigma_j$9, where $c$0. PH-equivalence consists of row and column permutations, multiplication by a nonzero Hadamard factor, and possibly transpose. For central $c$1, the classification simplifies sharply. Either $c$2 is rigid, meaning that its row and column equivalence relations are trivial, or

$c$3

for some $c$4, in which case $c$5 is H-equivalent to the signature and hence determinant-like (Pazzis, 2018).

The permanent provides the contrasting rigid example. For $c$6 and $c$7, the permanent-preserving maps are exactly

$c$8

with $c$9 and $p$0 a rank $p$1 matrix whose diagonal product is $p$2. Unlike determinant preservers, arbitrary $p$3 do not appear; only permutation matrices and normalized rank-1 Hadamard factors survive (Pazzis, 2018). A common misconception is therefore that “sign-like” invariants always admit the same symmetry group as the determinant. The central-map classification shows the contrary: determinant-type central functionals and rigid central functionals behave differently at the preserver level.

3. Sign regularity, SR/SSR signatures, and their preservers

A matrix $p$4 is strictly sign regular of order $p$5, written SSR$p$6, if for each $p$7 there exists $p$8 such that every $p$9 minor of $c+p=d$0 is nonzero and has sign $c+p=d$1. It is sign regular of order $c+p=d$2, written SR$c+p=d$3, if every nonzero $c+p=d$4 minor has sign $c+p=d$5, zeros being allowed. For $c+p=d$6, one gets the global notions SR and SSR. The associated signature is

$c+p=d$7

with $c+p=d$8 by convention. Totally positive and totally nonnegative matrices are exactly the cases with all $c+p=d$9 (Choudhury et al., 2024).

For surjective linear maps $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$0, the unconstrained SR/SSR-preserver classification is rigid. When either $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$1 or $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$2 with $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$3, $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$4 maps SR onto itself if and only if it maps SR$pos\le c,\qquad c-pos\equiv 0 \pmod 2,$5 onto itself, and this happens exactly when $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$6 is a composition of positive diagonal left and right multiplications, a global sign flip, row reversal by the exchange matrix $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$7, column reversal by $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$8, and, in the square case, transpose. For fixed signature $pos\le c,\qquad c-pos\equiv 0 \pmod 2,$9, the classification becomes stricter: only positive diagonal equivalences, simultaneous reversal $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$00, and, if $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$01, transpose preserve SR$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$02 or SSR$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$03 (Choudhury et al., 2024). The equivalence between preserving SR and preserving only SR$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$04 is a local-to-global principle: preservation of $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$05 and $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$06 minor signs already controls all orders.

A complementary entrywise theory studies functions $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$07 with $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$08 preserving SR or SSR. For fixed sign pattern $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$09, $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$10, the classification depends on $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$11 and on whether $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$12. On $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$13, SR$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$14-preservers are, according to dimension, either nonnegative functions ($$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$15), scaled signum or power functions $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$16 and $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$17 ($$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$18), pattern-dependent power/signum families ($$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$19), or only the cases $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$20 for $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$21. For SSR$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$22 on $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$23, the classification is stricter: $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$24 with $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$25 for $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$26, pattern-dependent ranges $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$27 or $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$28 for $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$29, and only $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$30, $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$31, for $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$32 (Choudhury et al., 22 Sep 2025).

When the sign pattern is not fixed, the allowable entrywise preservers change again. For all SR matrices, the high-dimensional regime $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$33 admits only nonzero constants or piecewise linear maps

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$34

whereas for all SSR matrices and $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$35, only the nonvanishing piecewise-linear form survives on $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$36 (Choudhury et al., 22 Sep 2025). This suggests that sign regularity is markedly more rigid under entrywise calculus than positive semidefiniteness.

4. Entrywise definiteness preservers and fixed-dimension positivity preservers

For entrywise transforms on Hermitian matrices, a sign preserver is a function $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$37 such that $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$38 is positive semidefinite or positive definite if and only if $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$39 is. In fixed dimension $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$40, the classification is complete over $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$41 and $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$42: the positive semidefinite and positive definite sign preservers are exactly positive multiples of continuous field automorphisms. Concretely,

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$43

over $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$44, and

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$45

over $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$46. The $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$47 case is exceptional: there the sign preservers are extensions of power functions, with $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$48 on the real line and $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$49 off the real axis, subject to conjugation symmetry (Guillot et al., 8 Jul 2025). On graph-constrained matrix spaces $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$50, the same dichotomy reappears: for trees, the class is larger, but for graphs containing a cycle, only the scaled identity or scaled conjugation remains (Guillot et al., 8 Jul 2025).

This rigidity contrasts with ordinary entrywise positivity preservation on $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$51, where fixed dimension allows much richer sign behavior in the Maclaurin coefficients. If

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$52

preserves positivity on $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$53, then the first $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$54 nonzero coefficients must be positive, but after those first $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$55 there is no further restriction on signs. On $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$56, the first $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$57 and last $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$58 nonzero coefficients must be positive, and any negative coefficient must be followed by at least $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$59 positive coefficients; these constraints are sharp (Khare et al., 2017). Thus fixed-dimensional positivity preservers can have highly nontrivial coefficient sign patterns even though fixed-dimensional sign preservers of definiteness are almost rigid.

A further variant forbids the entrywise function from acting on specified diagonal or principal blocks. If $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$60 is a partition of $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$61 with a uniformly bounded number $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$62 of blocks and some block of size at least $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$63, then dimension-free positivity preservers can be of the form

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$64

These provide the first examples of dimension-free entrywise positivity preservers that are not absolutely monotonic. By contrast, when the forbidden blocks overlap, one is forced back to the Herz-absolutely-monotone regime, and in the single-function formulation only the identity survives (Vishwakarma, 2020). A plausible implication is that block structure can interpolate between classical Schoenberg–Rudin rigidity and genuinely new preserver classes.

5. Positivity-preserving semigroups and fourth-order elliptic sign-preserving operators

On $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$65, a constant-coefficient operator

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$66

is a positivity preserver when

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$67

Such operators are characterized by moment sequences: $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$68 must be a multidimensional moment sequence, equivalently there exists a nonnegative Borel measure $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$69 with

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$70

Within the Fréchet Lie group $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$71 of constant-coefficient operators, a positivity preserver has a generator $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$72 with $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$73 if and only if its representing measure is infinitely divisible. The generators are precisely the Lévy–Khinchin ones: $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$74 with coefficients determined by a drift $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$75, a positive semidefinite covariance matrix $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$76, and a measure $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$77 satisfying the stated moment integrability conditions. In PDE language, these are exactly the generators of Lévy-type positivity-preserving semigroups on polynomials (Dio, 2023).

A different sign-preserving phenomenon occurs for fourth-order elliptic equations. In one dimension, if a fourth-order operator factors as

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$78

with

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$79

then any solution $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$80 of

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$81

satisfies either $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$82 or $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$83 on $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$84. The same sign-preserving conclusion holds for radially symmetric solutions in a ball and in an annulus with clamped Dirichlet conditions (Laurencot et al., 2013). The mechanism is a factorization-based replacement for a fourth-order maximum principle: the second-order factors satisfy standard maximum-principle hypotheses, and their combination yields negativity of the fourth-order solution for non-positive data.

These sign-preserving elliptic operators feed into spectral and nonlinear applications. The resolvent becomes strongly positive in appropriate cones, enabling Krein–Rutman arguments for positive eigenfunctions and simple principal eigenvalues. The same framework is used for nonlinear fourth-order equations, radially symmetric MEMS models, and Moreau-type polar-cone decompositions (Laurencot et al., 2013). This suggests a broad operator-theoretic notion of sign preservation: not merely invariance of a sign, but positivity-improving order structure after inversion or semigroup evolution.

6. Discrete sign properties and sign-pattern invariants

In entropy-stable finite difference schemes for scalar conservation laws, the relevant sign-preserver is a reconstruction procedure. If $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$85 is the entropy variable and

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$86

then entropy stability of the TeCNO flux

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$87

is ensured provided the reconstruction satisfies the sign property

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$88

The third-order SP-WENO reconstruction is built exactly to satisfy this property. Its weights are chosen from case-by-case feasible regions in the $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$89-plane so as to enforce convexity, mirror symmetry, the sign property, and third-order accuracy. The resulting reconstructed jumps satisfy the quantitative bound

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$90

the same bounding constant as ENO-2 and smaller than ENO-3 (Fjordholm et al., 2015). In this discrete setting, a sign preserver is therefore a nonlinear reconstruction whose interface jump cannot reverse the sign of the cell jump.

A complementary invariant-based viewpoint appears in the theory of sign patterns and sepr-sequences. For a real matrix $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$91, the sepr-sequence $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$92 records, for each order $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$93, whether the principal $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$94 minors are all positive, all negative, all zero, or occur with mixed sign/zero patterns. For a sign pattern $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$95, the sepr-set

$$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$96

collects the sepr-sequences of all realizations in its qualitative class. A sign pattern has a unique sepr-sequence if $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$97. A sufficient condition for uniqueness is that, for each $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$98, either every $$\sigma=\bigl(+,\operatorname{sgn}(a_{d-1}),\ldots,\operatorname{sgn}(a_0)\bigr).$$99 principal subpattern has a signed determinant, or there exist three such principal subpatterns with signed determinants $j$00, $j$01, and $j$02 (Hogben et al., 2018). Sign semi-stable patterns automatically satisfy the signed-determinant condition, hence have unique sepr-sequences; moreover, their nonzero order-$j$03 principal minors have sign $j$04, so the sequence is strongly parity-structured (Hogben et al., 2018).

This invariant language is not itself a preserver theory, but it clarifies what many sign preservers are actually preserving: not a scalar sign alone, but a stratified sign geometry of minors, determinants, jumps, or cones. Across these settings, the preserved object is typically rigid enough that once a small amount of local structure is fixed—$j$05 minors for SR, flattening minors for secants, determinant zero sets for Schur functionals, or interface jump signs for TeCNO schemes—the entire admissible transformation class becomes sharply constrained.