Mixed Segre Zeta Function

• Mixed Segre zeta function is a multivariate formal power series that encodes the degrees of mixed Segre classes for sequences of homogeneous ideals in projective spaces.
• It generalizes classical constructions by uniting Kleiman–Thorup’s mixed Segre classes and Aluffi’s one-variable Segre zeta function with a rational formulation controlled by generator degrees.
• The function is invariant under integral closure and produces modified numerators whose homogenizations are denormalized Lorentzian, ensuring log-concavity and deep links to convex geometry.

Searching arXiv for the main paper and closely related Segre zeta function work. The mixed Segre zeta function is a multivariate formal power series attached to a sequence of homogeneous ideals $I_1,\dots,I_m\subset k[x_0,\dots,x_n]$, defined so as to encode the degrees of the mixed Segre classes of the corresponding subschemes after extension to projective spaces of arbitrarily large dimension. In the formulation introduced in “Mixed Segre zeta functions and their log-concavity” (Cid-Ruiz, 8 Jul 2025), it simultaneously packages the push-forwards of all mixed Segre classes of all types $(i_1,\dots,i_m)$, generalizes both Kleiman–Thorup’s mixed Segre classes and Aluffi’s Segre zeta function, is rational with poles controlled by generator degrees, depends only on the integral closures of the ideals, and yields, after a natural modification, numerators whose homogenizations are denormalized Lorentzian.

1. Definition and algebro-geometric framework

Let $R=k[x_0,\dots,x_n]$ be the standard graded polynomial ring, let $I_1,\dots,I_m\subset R$ be homogeneous ideals, and let $Z_i\subset \mathbb{P}^n_k$ be the closed subschemes they define. If $Z\subset \mathbb{P}^n_k$ is the closed subscheme defined by the product ideal $I_1\cdots I_m$, then the mixed Segre zeta function is built from the mixed Segre classes of the $Z_i$ relative to $\mathbb{P}^n_k$ and their extensions to higher projective spaces (Cid-Ruiz, 8 Jul 2025).

The mixed Segre classes used here are defined in the sense of Kleiman–Thorup, but for an arbitrary number $m$ of closed subschemes. For an equidimensional scheme $(i_1,\dots,i_m)$0 of finite type over $(i_1,\dots,i_m)$1, with closed subschemes $(i_1,\dots,i_m)$2 and ideal sheaves $(i_1,\dots,i_m)$3, one forms the relative affine line

$(i_1,\dots,i_m)$4

the ideals $(i_1,\dots,i_m)$5, the associated multi-Rees algebra, and the resulting multi-Proj. Restricting to the divisor $(i_1,\dots,i_m)$6 and then to the subscheme $(i_1,\dots,i_m)$7, one obtains the Kleiman–Thorup transform $(i_1,\dots,i_m)$8, together with tautological line bundles $(i_1,\dots,i_m)$9. The mixed Segre class of type $R=k[x_0,\dots,x_n]$0 is then

$R=k[x_0,\dots,x_n]$1

and the total mixed Segre class is the corresponding generating polynomial in $R=k[x_0,\dots,x_n]$2.

To pass from mixed Segre classes to a zeta function, each ideal $R=k[x_0,\dots,x_n]$3 is extended to

$R=k[x_0,\dots,x_n]$4

producing subschemes $R=k[x_0,\dots,x_n]$5, and the product ideal defines $R=k[x_0,\dots,x_n]$6. If $R=k[x_0,\dots,x_n]$7 is the hyperplane class, a key stabilization statement is that for each multi-index $R=k[x_0,\dots,x_n]$8 with $R=k[x_0,\dots,x_n]$9, there exists an integer $I_1,\dots,I_m\subset R$0 such that

$I_1,\dots,I_m\subset R$1

and for fixed $I_1,\dots,I_m\subset R$2 the coefficient $I_1,\dots,I_m\subset R$3 is independent of $I_1,\dots,I_m\subset R$4 for $I_1,\dots,I_m\subset R$5 large enough. The mixed Segre zeta function is therefore defined by

$I_1,\dots,I_m\subset R$6

The coefficients thus encode, simultaneously for all ambient dimensions, the degrees of the push-forward mixed Segre classes.

When $I_1,\dots,I_m\subset R$7, the construction recovers the classical Segre class and Aluffi’s one-variable Segre zeta function. This is not merely an analogy: the mixed construction is designed so that the single-ideal case is literally its $I_1,\dots,I_m\subset R$8 specialization.

2. Relation to classical Segre zeta functions and earlier theories

The mixed Segre zeta function unifies two previously distinct directions. On one side are Kleiman–Thorup’s mixed Segre classes, originally developed for two closed subschemes and extended here to arbitrary $I_1,\dots,I_m\subset R$9 through the joint blow-up and multi-Rees formalism. On the other side is Aluffi’s Segre zeta function, which packages the Segre classes of a single homogeneous ideal across all higher projective cones into a rational one-variable series (Aluffi, 2016).

In the case $Z_i\subset \mathbb{P}^n_k$0, the mixed Segre zeta function reduces to

$Z_i\subset \mathbb{P}^n_k$1

where $Z_i\subset \mathbb{P}^n_k$2 is determined by

$Z_i\subset \mathbb{P}^n_k$3

This is exactly Aluffi’s Segre zeta function. The mixed theory therefore generalizes the rationality, generator-degree control, and integral-closure invariance of the one-variable setting, while adding genuinely multivariate structure and a Lorentzian/log-concavity theorem absent from the original one-ideal formulation (Cid-Ruiz, 8 Jul 2025).

A central structural relation is the “mixed formula.” If

$Z_i\subset \mathbb{P}^n_k$4

one defines a modified series

$Z_i\subset \mathbb{P}^n_k$5

Then

$Z_i\subset \mathbb{P}^n_k$6

Geometrically, this reflects the identity

$Z_i\subset \mathbb{P}^n_k$7

A frequent misconception is to identify the mixed Segre zeta function with the Segre zeta function of the product ideal. The mixed formula shows that the product-ideal zeta is recovered only after a factorial reweighting and diagonal specialization; the mixed object contains strictly finer multigraded information.

The relative and multigraded literature provides nearby constructions rather than direct substitutes. The relative Segre zeta function for subschemes of projective bundles and the two-variable Segre zeta function for products of projective spaces show that Segre-zeta formalism naturally admits multivariable extensions, but those constructions organize ambient multigrading rather than mixed Segre classes of several ideals in the Kleiman–Thorup sense (Jorgenson, 2019). This suggests that “mixed” has at least two distinct meanings in the Segre-zeta literature: mixed ideals and mixed ambient gradings.

3. Rationality and the structure of poles

Let

$Z_i\subset \mathbb{P}^n_k$8

The main algebraic theorem states that the mixed Segre zeta function is rational: $Z_i\subset \mathbb{P}^n_k$9 where $Z\subset \mathbb{P}^n_k$0 is a polynomial with nonnegative integer coefficients (Cid-Ruiz, 8 Jul 2025). Accordingly, the only possible poles in the variable $Z\subset \mathbb{P}^n_k$1 occur at $Z\subset \mathbb{P}^n_k$2, so each variable detects only the degrees of generators of the corresponding ideal $Z\subset \mathbb{P}^n_k$3.

The proof adapts Aluffi’s “shadow of the blow-up” method to the mixed setting. The construction uses vector bundles on $Z\subset \mathbb{P}^n_k$4 whose sections reproduce the chosen generators, identifies the joint blow-up with the image of a rational map to a product of projective bundles, and then combines an explicit blow-up formula for mixed Segre classes with functoriality under projective projections $Z\subset \mathbb{P}^n_k$5. A key truncation lemma shows that, once $Z\subset \mathbb{P}^n_k$6 is sufficiently large, the low-degree part of the zeta function is controlled by finitely many data involving the generator degrees and the mixed Segre classes in $Z\subset \mathbb{P}^n_k$7.

The denominator is not merely formal bookkeeping. It expresses a separation of variables intrinsic to the mixed theory: $Z\subset \mathbb{P}^n_k$8 couples only to the degree data of $Z\subset \mathbb{P}^n_k$9. This makes the mixed rationality theorem a direct multivariate analogue of Aluffi’s denominator theorem for a single ideal, but with the notable strengthening that the denominator factors by ideal rather than only after collapsing all data to one parameter (Aluffi, 2016).

Not every potential pole survives in the reduced denominator. The paper emphasizes that some factors disappear after passage to integral closure, which shows that the displayed denominator is an a priori one rather than a canonical minimal factorization. This is one place where the rational form reflects generating degrees but the actual zeta function reflects deeper Rees-theoretic data.

4. Integral closure, invariance, and integral dependence

A defining structural property of the mixed Segre zeta function is its invariance under integral closure. If $I_1\cdots I_m$0 denotes the integral closure of $I_1\cdots I_m$1, then

$I_1\cdots I_m$2

The proof uses the existence of a finite birational morphism between the joint blow-ups attached to $I_1\cdots I_m$3 and to $I_1\cdots I_m$4, together with an expression of mixed Segre classes in terms of intersections of powers of the exceptional divisors and the functoriality of mixed Segre classes under proper push-forward and flat pull-back (Cid-Ruiz, 8 Jul 2025).

This result extends a standard feature of ordinary Segre classes and of Aluffi’s Segre zeta function: Segre data are insensitive to passing from an ideal to its integral closure. In the one-ideal case, this invariance is not only a structural property but also an exact criterion for integral dependence among homogeneous ideals: for homogeneous $I_1\cdots I_m$5,

$I_1\cdots I_m$6

as proved in “Segre classes and integral dependence” (Cid-Ruiz, 9 Dec 2025). The mixed paper does not formulate an analogous criterion for tuples of ideals, but the invariance theorem strongly suggests that mixed zeta functions are governed by the integral-closure classes of the individual ideals rather than by chosen generating sets.

This invariance also clarifies a common misunderstanding about denominator data. The poles are described by the degrees of homogeneous generators, yet the zeta function itself depends only on integral closures. Consequently, the actual pole structure is constrained both by the displayed generating degrees and by cancellations forced by integral dependence. In that sense, the mixed Segre zeta function simultaneously remembers presentation-level degree data and discards presentation-level ideal-theoretic redundancy.

5. Lorentzian structure and log-concavity

The second principal theorem concerns a modification of the rational expression. If

$I_1\cdots I_m$7

one defines $I_1\cdots I_m$8 by

$I_1\cdots I_m$9

Equivalently,

$Z_i$0

The polynomial $Z_i$1 has integer, indeed nonnegative, coefficients. If $Z_i$2 is algebraically closed and $Z_i$3 denotes the homogenization

$Z_i$4

then $Z_i$5 is denormalized Lorentzian (Cid-Ruiz, 8 Jul 2025).

Here “denormalized Lorentzian” means that after dividing each coefficient by the corresponding factorial $Z_i$6, the resulting homogeneous polynomial is Lorentzian in the sense of Brändén–Huh. The theorem is proved by rewriting $Z_i$7 as a push-forward of total Chern classes of certain globally generated vector bundles $Z_i$8 on the joint blow-up and then applying a general result asserting that generating functions built from push-forwards of Chern classes of globally generated bundles yield denormalized Lorentzian polynomials.

The significance of the theorem lies in the strong coefficient inequalities forced by the Lorentzian condition. Since the normalization $Z_i$9 is Lorentzian, its univariate specializations along nonnegative directions have log-concave coefficient sequences, and special slices such as $\mathbb{P}^n_k$0 or single-variable coordinate slices inherit log-concavity after normalization. The paper emphasizes that mixed Segre zeta functions therefore provide a systematic source of denormalized Lorentzian polynomials arising from arbitrary homogeneous ideals.

This result should be distinguished from an adjacent conjectural picture in the multihomogeneous setting. The paper places its theorem alongside an open conjecture of Aluffi asserting that for a multihomogeneous ideal defining a subscheme of a product of projective spaces, the homogenization of the numerator of $\mathbb{P}^n_k$1 should be denormalized Lorentzian. The mixed theorem gives a positive answer in a closely related but different context: here the relevant numerators arise from volume polynomials, which are known to be Lorentzian. This suggests a deep connection between Segre-zeta constructions, volume-type positivity, and combinatorial log-concavity, but the two settings are not identical.

6. Geometric interpretation, computation, examples, and related directions

The coefficients of the mixed Segre zeta function are intersection-theoretic invariants on the joint blow-up. For $\mathbb{P}^n_k$2, the blow-up formula gives

$\mathbb{P}^n_k$3

where $\mathbb{P}^n_k$4 is the joint blow-up and the $\mathbb{P}^n_k$5 are its exceptional divisors. The zeta coefficients are therefore degrees of intersection cycles on $\mathbb{P}^n_k$6, and the mixed Segre zeta function may be viewed as an invariant of the multi-Rees geometry of the tuple $\mathbb{P}^n_k$7 (Cid-Ruiz, 8 Jul 2025).

The paper also gives a computable expression in terms of projective degrees of the rational map

$\mathbb{P}^n_k$8

determined by the generators. This yields an algorithm for computing mixed Segre classes, and a Macaulay2 implementation is reported. In this respect the mixed theory extends the computational role already played by the classical Segre zeta function, which was introduced in part to accelerate computations of Segre classes and later connected to Newton–Okounkov convex geometry in the one-ideal case (Aluffi, 2018).

Several examples illustrate the general formulas. For two disjoint complete intersections

$\mathbb{P}^n_k$9

in disjoint sets of variables, with degrees $m$0 and $m$1, one has

$m$2

and

$m$3

For a more concrete example, if $m$4 and $m$5, then for the product ideal $m$6,

$m$7

Here the numerator of $m$8 is $m$9, whose homogenization $(i_1,\dots,i_m)$00 is Lorentzian. A bivariate example gives

$(i_1,\dots,i_m)$01

with homogenization

$(i_1,\dots,i_m)$02

again Lorentzian.

The broader context makes clear that mixed Segre zeta functions occupy an intersection of several active lines of research. Aluffi’s original Segre zeta function established rationality, positivity, and integral-closure invariance for a single ideal (Aluffi, 2016). Newton–Okounkov methods later gave a convex-geometric integral formula for $(i_1,\dots,i_m)$03 in the one-variable case (Aluffi, 2018). Relative and product versions showed that Segre-zeta formalism extends naturally to projective bundles and products of projective spaces (Jorgenson, 2019). The mixed theory of (Cid-Ruiz, 8 Jul 2025) brings these strands together around joint blow-ups, multi-Rees algebras, and Lorentzian positivity. A plausible implication is that mixed Segre zeta functions provide a natural framework for further work at the interface of intersection theory, integral closure, convex geometry, and log-concavity.