The symmetric functions catalog

#Interlacing and interleaving roots

Interlacing is one of the most flexible ways to certify real-rootedness. This page separates four recurring viewpoints:

• root-level interlacing, Sturm sequences, and derivative recurrences;

• matrix certificates such as Bézout matrices, Hurwitz matrices, and Lace matrices;

• compatible polynomials and common interlacings;

• operators and decompositions that preserve real-rootedness.

Coefficient-level total-positivity tools are collected on the Pólya frequency page, while the multivariate operator viewpoint is covered on the stable-polynomial page.

We first fix the interlacing convention used throughout, following [Wag92]; see also [Br15].

Wagner’s lemma is one of the basic closure tools for interlacing induction.

D.G. Wagner also shows that \(f \interl g\) implies that \(f \interl \lambda f + \mu g \interl g\) for all \(\lambda, \mu \geq 0.\)

As an example, if \(f\) is real-rooted, then \(f' \interl f.\) Hence, recursions involving the derivative play well with interlacing roots. The differential recurrence for Eulerian polynomials is a classical example: it produces a strict Sturm sequence. The Eulerian root argument records the sign analysis. The next three criteria cover derivative recurrences, two-term recurrences, and affine combinations, respectively. Vector recurrences are treated by the matrix criterion below.

A useful derivative criterion is due to S.-M. Ma and Y. Wang [MW08, Thm. 2].

The theorem states more regarding root placements and multiplicities.

Favard recurrences give a second standard interlacing source, especially for orthogonal polynomial families.

The following criterion from [LW07, Thm. 2.1] is often useful.

The following affine-coefficient criterion is useful when neither summand is a derivative.

By applying this criterion to row-generating polynomials, Wang and Yeh also give conditions ensuring that bilinear triangular recurrences produce Pólya frequency sequences.

#Sturm sequences

There are two related uses of the term Sturm sequence. In enumerative combinatorics, it is fairly common to see sequences of polynomials \(\{ P_n(t) \}_{n \geq 0}\) with the property that \(P_j(t) \interl P_{j+1}(t)\) for all \(j.\) If the leading coefficient is always positive and \(\deg(P_n)=n,\) this is called a Sturm sequence, see [LW07]. If we drop the degree condition, the sequence is called a generalized Sturm sequence.

The notion of Sturm-unimodality is defined in the analogous manner.

There are many Sturm sequences on permutations and Catalan objects.

The classical Sturm sequence, or Sturm chain, instead starts from one polynomial \(f.\) Set \(S_0=f,\) \(S_1=f',\) and recursively take the negative remainders \[S_{j+1}=-\operatorname{rem}(S_{j-1},S_j)\] until the remainder is zero. For a real number \(x,\) let \(V_f(x)\) be the number of sign changes in \[S_0(x),S_1(x),\dotsc\] after zero entries are omitted.

In particular, evaluating the signs at \(-\infty\) and \(+\infty\) gives the total number of distinct real zeros. This provides an exact coefficient-based test for real-rootedness: if \(g\) is the squarefree part of \(f,\) then \(f\) is real-rooted if and only if Sturm’s theorem counts \(\deg(g)\) distinct real zeros of \(g.\)

#Bézout matrices and Hurwitz–Lace

For arbitrary polynomials \(f(t)\) and \(g(t)\) with \(\deg(f) = \deg(g)+1,\) let \(B(f,g)\) be the Bézout matrix, whose \((i,j)\) entry is equal to the coefficient of \(x^{i-1} y^{j-1}\) in the polynomial \[\frac{f(x)g(y) - f(y)g(x)}{x-y}.\]

This criterion packages interlacing roots as total nonnegativity of a structured Hurwitz matrix.

The converse fails even when every coefficient is positive. For example, \[(t+3)(t^2-2t+10)=t^3+t^2+4t+30\] has positive coefficients but has the two zeros \(1\pm3i\) in the open right half-plane. Thus Hurwitz stability is a sufficient certificate for coefficient positivity, but is strictly stronger than coefficient positivity.

Thus the Hermite–Biehler and Routh–Hurwitz criteria also translate root-location questions into interlacing and total nonnegativity. The block-Toeplitz and network viewpoint is discussed in [Ang23, Sec. 4], and weak/quasi-stable boundary cases are treated in [AGT17].

#Interlacing sequences and matrix criteria

The following definitions are from [Br15].

P. Brändén gives useful examples of how interlacing sequences can be used to prove real-rootedness [Br15]. In particular, rook polynomials associated with Ferrers diagrams can be proved to be real-rooted using this method.

The interlacing hypothesis is essential: an arbitrary positive sum of real-rooted polynomials need not be real-rooted. A useful specialization is that the barycentric subdivision of every cubical complex admitting a stable shelling has a real-rooted \(h\)-polynomial [HS22, Thm. 4.7].

#Matrices preserving interlacing sequences

The following matrix criterion is proved in [Br15, Thm. 7.8.5].

For a constant matrix, total non-negativity gives a simpler sufficient test.

Thus a coefficient matrix supplied by a planar network can be used directly to mix an interlacing refinement.

[
  [[0, 1], [0, t]],
  [[0, 1], [t, 0]],
  [[1, 0], [0, 1]],
  [[1, 0], [t, 0]],
  [[t, 0], [0, t]],
  [[0, 1], [t, 1]],
  [[0, 1], [t, t]],
  [[1, 0], [1, 1]],
  [[1, 0], [t, 1]],
  [[1, 1], [0, 1]],
  [[1, 1], [t, 0]],
  [[t, 0], [t, t]],
  [[t, 1], [0, t]],
  [[t, 1], [t, 0]],
  [[t, t], [0, t]],
  [[1, 1], [t, 1]],
  [[1, 1], [t, t]],
  [[t, 1], [t, t]]
]

#Compatible polynomials

Chudnovsky and Seymour introduced a concept closely related to interleaving sequences of polynomials. The polynomials \(f_1,\dotsc,f_k \in \setR[z]\) have a common interlacing if some \(h \in \setR[z]\) exists with \(h \interl f_j\) for all \(j.\)

This signed pencil criterion is the two-polynomial counterpart of the nonnegative compatibility theorem below.

L. L. Liu studies this compatible-zero method further in [Liu12]. In particular, she gives a necessary and sufficient criterion for two real-rooted polynomials with opposite signs of their leading coefficients to be compatible, and a direct induction proof of the pairwise-compatible positive-leading-coefficient case. Her criterion is a root-count condition. For nonconstant \(f\) and \(g,\) the correct statement is that \(f\) and \(g\) are compatible if and only if either this condition holds, or they share a zero \(r\) such that \(f/(x-r)\) and \(g/(x-r)\) are compatible. The second alternative is missing from the published statement, and it is needed: \(x\) and \(-x^2\) are compatible but fail the root-count condition.

Thus mixed signs require a triple test rather than the pairwise test in the positive-leading-coefficient case.

Another useful technique is presented in [Ben18], where Christoffel–Darboux type identities are used to show interlacing. This exploits an idea similar to Bézout matrices. By taking the limit, \[\lim_{y\to x} \frac{f(x)g(y)-f(y)g(x)}{x-y} = f'(x)g(x) - g'(x)f(x)\] which is the Wronskian of \(f\) and \(g.\)

The multiplicity condition cannot be dropped: \(f=(x-1)^3\) and \(g=x^3\) have equal degree and the nonnegative Wronskian \(3x^2(x-1)^2,\) but \(x^3\) does not interleave \((x-1)^3.\)

Thus one way to prove an interlacing statement is to show that this Wronskian is nonnegative, perhaps by rewriting it as a sum of squares. This is especially useful when a Christoffel–Darboux identity gives a closed form for the difference quotient above.

For example, if \(A\) is a real symmetric matrix and \(A_S\) denotes the principal submatrix obtained by deleting the rows and columns indexed by \(S,\) then the characteristic polynomials \(\{\det(xI-A_S): |S|=k\}\) form an interlacing family. The root polynomial is the \(k\)th derivative of the characteristic polynomial of \(A.\) This refines the Cauchy interlacing theorem from a single principal submatrix to a whole tree of principal submatrices.

The method was introduced by A. Marcus, D. Spielman, and N. Srivastava for graph signings. The Godsil–Gutman identity says that the expected characteristic polynomial of a uniformly random signed adjacency matrix is the matching polynomial. They proved that the signed characteristic polynomials form an interlacing family. Combined with the matching-polynomial root bound, this selects a signing whose largest eigenvalue is at most \(2\sqrt{d-1}\) for a graph of maximum degree \(d.\) Applied repeatedly to \(2\)-lifts of bipartite graphs, this gives infinite families of bipartite Ramanujan graphs of every degree [MSS15].

Their sequel combines interlacing families with mixed characteristic polynomials and barrier functions; this yields a positive solution of the Kadison–Singer problem [MSS15].

#Fully interlacing sequences and the \(\operatorname{Lace}\) matrix

A stronger coefficient-level version of interlacing was introduced by C. A. Athanasiadis and D. G. Wagner [AW24].

For a column vector \(A=(A_0,A_1,\dotsc,A_{p-1})^T,\) Athanasiadis–Wagner call the sequence fully interlacing if \(\operatorname{Lace}(A)\) is totally nonnegative. Thus every finite minor of this infinite coefficient-convolution matrix is nonnegative.

Full interlacing implies ordinary pairwise interlacing, but the converse is false. The distinction is useful: pairwise interlacing is a root-level condition, whereas full interlacing is a total-nonnegativity condition on a structured coefficient matrix.

Whenever the product \(AB\) is defined, \[\operatorname{Lace}(AB)=\operatorname{Lace}(A)\operatorname{Lace}(B).\] This is convenient, since totally nonnegative matrices are closed under multiplication.

#Preserving operators and symmetric decompositions

The previous sections treat interlacing as a certificate. The next examples show how the certificate is produced by structural operations: symmetric decompositions in Ehrhart theory, differential operators, products of Ehrhart functions, and basis transformations such as the derangement and Narayana transformations.

#Symmetric decompositions

The following setup from [BS18] is especially useful for \(h\)-polynomials arising in Ehrhart theory.

Given \(p \in \setR[x]\) with \(\deg(p) \leq d,\) define \[I_d(p)(x) \coloneqq x^d p(1/x).\] Then unique polynomials \(a,b \in \setR[x]\) exist such that \[p(x) = a(x) + x b(x), \qquad I_d(a)=a, \qquad I_{d-1}(b)=b.\] This is the symmetric \(I_d\)-decomposition of \(p,\) and it is given explicitly by \[a = \frac{p - x I_d(p)}{1-x}, \qquad b = \frac{I_d(p)-p}{1-x}.\]

It is also convenient to associate to \(p\) the polynomial \[f(x) \coloneqq (1+x)^d p\!\left( \frac{x}{1+x}\right),\] and to define \[R_d(f)(x) \coloneqq (-1)^d f(-1-x).\] The paper uses the proper-position relation \(\prec\); in the present nonnegative coefficient setting, this can be read as an interlacing relation.

Thus, an interlacing statement between a polynomial and its reciprocal forces the symmetric pieces of the decomposition to be real-rooted as well. This is one reason the theorem shows up naturally for reflexive \(h^*\)-polynomials; see magic positivity on the polytope page.

For integers \(r\geq1\) and \(d\geq0,\) define the Veronese numerator operator \(U_r^{d+1}\) by \[\sum_{n\geq0}a_{rn}t^n =\frac{U_r^{d+1}h(t)}{(1-t)^{d+1}} \qquad\text{whenever}\qquad \sum_{n\geq0}a_nt^n=\frac{h(t)}{(1-t)^{d+1}}.\]

For a \(d\)-dimensional lattice polytope, \(U_r^{d+1}h_P^*=h_{rP}^*.\) Consequently every such dilation in the stated range has a real-rooted symmetric decomposition; it is interlacing when \(P\) is Gorenstein [Joc22, Cor. 6.1–6.2].

There is a parallel tool for colored barycentric subdivisions. Given \(h(x)=\sum_{i=0}^n c_ix^i,\) set \[f(x)=\sum_{i=0}^n c_ix^i(1+x)^{n-i}, \qquad f(0)+\sum_{m\geq1}\bigl(f(rm)-f(rm-1)\bigr)x^m =\frac{D_{n,r}h(x)}{(1-x)^n}.\] For every \(r\geq1,\) \(D_{n,r}h\) is real-rooted whenever the \(c_i\) are nonnegative. If additionally \(\sum_{j=0}^i c_j\leq\sum_{j=0}^i c_{n-j}\) for \(0\leq i\leq\lfloor n/2\rfloor,\) its symmetric decomposition is also nonnegative and real-rooted [AT21, Thm. 1.3].

#Operators preserving real-rootedness

A multivariate approach to preserving real-rootedness is provided by stable polynomials, where linear operators preserving stability imply real-rootedness of specializations.

D.G. Wagner proves the following result [Wag92].

In particular, this shows that if \(f\) and \(g\) are Ehrhart functions of two polytopes \(P_f\) and \(P_g\) with real-rooted \(h^*\)-polynomials, then the \(h^*\)-polynomial of the polytope product \(P_f * P_g\) is also real-rooted. For examples of polytopes with real-rooted \(h^*\)-polynomials, see tableaux and polytopes.

The order cannot be fixed in general. The operator \(p(x)\mapsto p(-x)\) preserves real-rootedness and \(x \interl x-1,\) but the normalized images satisfy \(x+1 \interl x\) rather than \(x \interl x+1.\)

#Schur–Szegő composition and interlacing

At a fixed degree bound, Schur–Szegő composition with a fixed polynomial is a diagonal operator, so the finite multiplier criterion applies. Kostov [Kos10] proves interlacing properties for polynomials arising from eigenvectors of the coefficient map defined by Schur–Szegő factorization.

For separable-permutation descent polynomials, Zhang compares two degree-dependent diagonal compression maps. He represents their outputs through one common finite Schur–Szegő multiplier and explicit kernels [Zha26, Thm. 4.1 and Lem. 4.2].

#Derangement and Narayana transformations

Another operator with strong interlacing properties is the following. Let \(d_n(x)\) denote the classical derangement polynomial, and define the derangement transformation \(D : \setR[x] \to \setR[x]\) by \[D(x^n) \coloneqq d_n(x).\] If \(p\) and \(q\) are degree-\(n\) polynomials with roots \(\alpha_1 \leq \dotsb \leq \alpha_n\) and \(\beta_1 \leq \dotsb \leq \beta_n\) in \([-1,0],\) write root order \(p \leq_{\mathrm{root}} q\) if \(\alpha_k \leq \beta_k\) for all \(k.\)

In particular, \(D(p)\) and \(D(q)\) are real-rooted and have interlacing zeros. As a consequence, if \[p(x)=\sum_{k=0}^n c_k x^k (x+1)^{n-k}, \qquad c_k \geq 0,\] then [BS18, Cor. 3.7] gives \[A_n \prec D(p) \prec d_n.\]

For \(0\leq k\leq n,\) define \[d_{n,k}(x) \coloneqq D\bigl(x^k(1+x)^{n-k}\bigr) =\sum_{\substack{\pi\in\symS_n\\ \operatorname{Fix}(\pi)\subseteq[n-k]}} x^{\exc(\pi)}.\] The excedance interpretation is due to C. A. Athanasiadis [Ath25, Prop. 4.1(a)]. Thus \(d_{n,0}(x)=A_n(x),\) while \(d_{n,n}(x)=d_n(x).\)

More generally, suppose that fixed points are forbidden on any specified set \(S\subseteq[n]\) with \(|S|=k.\) Fixed-point deletion preserves the excedance number, so inclusion-exclusion gives \[\sum_{\substack{\pi\in\symS_n\\ \operatorname{Fix}(\pi)\cap S=\varnothing}} x^{\exc(\pi)} =\sum_{T\subseteq S}(-1)^{|T|}A_{n-|T|}(x) =\sum_{i=0}^k(-1)^i\binom{k}{i}A_{n-i}(x) =d_{n,k}(x).\] Consequently, only the cardinality of the forbidden set matters.

The polynomials \(x^k(1+x)^{n-k}\) form a chain in root order. Hence, for every \(0\leq i\lt{}j\leq n,\) [BS18, Thm. 3.6 and Cor. 3.7] imply \(d_{n,i}(x)\prec d_{n,j}(x)\); in particular, \[d_{n,0}(x)\prec d_{n,1}(x)\prec\dotsb\prec d_{n,n}(x).\] Therefore \((d_{n,k}(x))_{0\leq k\leq n}\) is an interlacing sequence and every \(d_{n,k}(x)\) is real-rooted. The proof of [BS18, Lem. 3.5] already keeps separate parameters for the individual fixed points, giving a stronger fixed-point-weight refinement before specialization.

This is one route to the interlacing results for colored Eulerian and colored derangement polynomials on the permutation page.

There is an analogous basis-transformation viewpoint for other classical polynomial triangles. The Eulerian transformation \(A(x^n)=A_n(x)\) does not preserve real-rootedness in complete generality [BJ22], but it does preserve real-rootedness on the nonnegative cone discussed on the gamma-positivity page [Ath25]. In parallel, J. Mao and L. Wang prove that the Narayana transformation \(T_{N_m}(x^n)=N_{n,m}(x),\) interpolating between the type \(B\) and type \(A\) Narayana polynomials, preserves real-rootedness for polynomials with nonnegative coefficients; see Narayana transformations and [MW26].

L. L. Liu and X. Yan [LY25] refine this picture by studying the triangular matrices arising from binomial-type transforms. For the derangement transformation, for every fixed \(k\) the column \[d_{k,k}(x),\,d_{k+1,k}(x),\,d_{k+2,k}(x),\dotsc\] is a generalized Sturm sequence. Their recurrence framework also recovers the fixed-\(n\) row interlacing above. They prove analogous statements for the type \(B\) and \(r\)-colored derangement transformations, and obtain sufficient conditions for such row and column interlacing in a more general recurrence setup.

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