On the Sharpness of Khovanskii's Bezout-type Bound for Pfaffian Functions

Abstract: Khovanskii's theorem gives a Bezout-type upper bound for the number of isolated real solutions of a system of $n$ Pfaffian equations in $n$ variables in terms of three complexity parameters: the chain-degree $α$, the degrees $β<em>i$ of the Pfaffian functions, and the order $s$ of the underlying Pfaffian chain. Despite its fundamental role in Pfaffian geometry and o-minimality, little is known about the sharpness of this bound. We investigate the theorem from a parameter-by-parameter perspective. We show that its dependence on the chain-degree $α$ is asymptotically sharp by constructing, for every $α,s \in \mathbb{N}$, a Pfaffian function of format $(α,1,s)$ with at least $α<sup>s$ nondegenerate real zeros. We also show that its dependence on the degrees $β_i$ is asymptotically sharp: for fixed $n$ and $s$, we construct Pfaffian systems having $Ω</em>{n,s}(β<sup>{n+s})$ regular common zeros, matching the order of growth predicted by Khovanskii's theorem as $β\to\infty$.

• The paper proves that Khovanskii’s bound is asymptotically sharp in the chain-degree α by constructing format (α,1,s) Pfaffian functions with at least α^s distinct nondegenerate real zeros.
• The paper establishes β^{n+s}-scale sharpness in the equation degrees through an algebraic-independence assumption and a dimension-count argument producing many regular common zeros.
• The paper finds no examples matching the bound’s 2^{Θ(s²)} dependence on chain order, suggesting this factor may be improvable while leaving simultaneous α–β sharpness at fixed order unresolved.

Khovanskii's theorem is the Pfaffian analogue of Bézout's theorem: it bounds the number of isolated real solutions of a system of $n$ Pfaffian equations in $n$ variables in terms of three complexity parameters — the order $s$ of the underlying Pfaffian chain, its chain-degree $\alpha$, and the degrees $\beta_i$ of the equations. The bound has dominant growth of order $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$, yet prior to this work essentially nothing was known about whether these dependencies are tight. The paper under review (2606.24373) initiates a parameter-by-parameter sharpness analysis, proving that the bound is asymptotically sharp in both $\alpha$ and $\beta$, while presenting evidence that its dependence on $s$ is far from optimal.

Background: Pfaffian functions and Khovanskii's bound

A Pfaffian chain of order $s$ and chain-degree $n$0 on an open set $n$1 is a sequence of smooth functions $n$2 whose derivatives satisfy a triangular system of first-order differential equations with polynomial coefficients of degree at most $n$3. A Pfaffian function with respect to such a chain is a polynomial of degree at most $n$4 in the variables and the chain functions; one says it has format $n$5. The class includes polynomials, exponentials, iterated exponentials, restricted trigonometric functions, logarithms, and fewnomials (a polynomial with $n$6 monomials is Pfaffian of format $n$7). Pfaffian sets form an o-minimal structure by Speissegger's theorem, and quantitative control of their complexity underlies effective bounds on connected components, Betti numbers, and stratification complexity, as well as Wilkie's proof that the real exponential field is o-minimal.

For a fixed chain of chain-degree $n$8 and order $n$9, Khovanskii's theorem states that a system $s$0 of Pfaffian functions of degrees $s$1 has at most

$s$2

isolated real zeros. When $s$3 this recovers Bézout's inequality exactly. The paper asks whether each factor's growth rate can be attained by explicit or existential constructions.

Sharpness in the chain-degree $s$4

The first main result shows that the $s$5 growth predicted by the bound is attained: for every $s$6 there exists a Pfaffian function of format $s$7 with at least $s$8 nondegenerate real zeros. Since Khovanskii's bound for format $s$9 is itself of order $\alpha$0, this establishes asymptotic sharpness in $\alpha$1 for fixed $\alpha$2, $\alpha$3, and $\alpha$4.

The construction is analytic and iterative. One starts with $\alpha$5 and $\alpha$6, so that $\alpha$7 has format $\alpha$8. A careful intermediate-value argument locates exactly $\alpha$9 nondegenerate zeros of $\beta_i$0: one near $\beta_i$1, one at $\beta_i$2, and one within $\beta_i$3 of each integer $\beta_i$4, alternating to the left or right depending on parity; degeneracy is ruled out by contradiction arguments along sequences $\beta_i$5. For sufficiently large $\beta_i$6, one obtains pairwise disjoint compact intervals $\beta_i$7 around the roots $\beta_i$8 on which $\beta_i$9 is a local diffeomorphism and, crucially, whose images cover all the intervals: $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$0.

The full construction then builds $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$1 and considers $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$2, which has format $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$3 with respect to the chain $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$4. Each word $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$5 indexes a nested compact interval $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$6 obtained by pulling back along inverse branches; the covering property gives every interval $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$7 descendants, absence of critical points keeps them well-defined, and a homeomorphism bookkeeping argument shows that at each level $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$8 the $2^{\Theta(s^2)}\,\alpha^s\,(\max_i \beta_i)^{n+s}$9 intervals are pairwise disjoint. At level $\alpha$0, each interval satisfies $\alpha$1, so the intermediate value theorem yields a fixed point of $\alpha$2 in each interval — hence $\alpha$3 distinct zeros of $\alpha$4. Nondegeneracy follows from a chain-rule estimate: along any fixed point, $\alpha$5 for large $\alpha$6, where $\alpha$7 bounds $\alpha$8 away from zero on the intervals and $\alpha$9 bounds them away from the origin.

An immediate consequence is a lower barrier: no improvement of Khovanskii's bound can grow as $\beta$0 when $\beta$1 are fixed, since the construction already forces $\beta$2 zeros. Any sharpening must therefore target the $\beta$3 prefactor rather than the exponential-in-$\beta$4 behavior itself.

Sharpness in the degrees $\beta$5

The second main result addresses the regime where $\beta$6, $\beta$7, and $\beta$8 are fixed and $\beta$9, where Khovanskii's bound grows like $s$0. Assuming the existence of a Pfaffian chain of order $s$1 and chain-degree $s$2 that is algebraically independent over $s$3, the paper proves that for every $s$4 there exist $s$5 distinct Pfaffian functions of format $s$6 sharing at least

$s$7

distinct regular common real zeros. This matches the $s$8 growth predicted by the bound.

The proof is a dimension count. Under algebraic independence, the space $s$9 spanned by monomials $s$0 of total degree at most $s$1 has dimension $s$2. Imposing $s$3 prescribed common zeros together with prescribed derivative data ($s$4 affine-linear conditions total) leaves a nonempty solution set in $s$5 whenever $s$6, and the derivative conditions force each point to be a regular zero of the resulting system.

To instantiate the hypothesis, the paper verifies that the iterated-exponential chain $s$7, $s$8 is algebraically independent over $s$9. The proof is an asymptotic dominance argument: if a nonzero polynomial relation existed, dividing by the term with $n$00-maximal multi-index and evaluating along lines $n$01 shows each subdominant summand decays faster than any polynomial grows, forcing the leading coefficient polynomial to vanish identically — a contradiction. Combining this with the dimension count yields the corollary that for fixed $n$02 and $n$03 there are systems of format $n$04 with $n$05 regular common zeros.

Combined regimes and the dependence on $n$06

Because the $n$07-construction lives in one variable and the $n$08-construction uses $n$09 variables, taking Cartesian products gives, for every $n$10, a system of format $n$11 with $n$12 regular zeros. The cost is a doubling of the chain order from $n$13 to $n$14; whether a family of order-$n$15 systems achieving $n$16 exists is left open.

In contrast, the evidence regarding $n$17 points away from sharpness. Even at small parameters the gap is large: $n$18 has format $n$19, for which Khovanskii's bound gives $n$20, while the actual number of zeros is $n$21. The authors report substantial difficulty finding examples approaching the bound even for chains of length two, and note (via personal communication with Vorobjov) the expectation that a bound exponential in $n$22 rather than $n$23 may hold. Two partial improvements are recorded. For fewnomials, Bihan–Rojas–Sottile improved the $n$24 dependence. For univariate polynomials in $n$25, a result of Barbagallo–Jeronimo–Sabia gives at most $n$26 real zeros for $n$27 with $n$28, $n$29 — linear in $n$30, versus the quadratic $n$31 suggested by the format $n$32. Consequently Khovanskii's bound can be sharp for such functions only when $n$33.

Limitations and open questions

Several caveats qualify the results. Theorem on $n$34-sharpness is conditional on the existence of an algebraically independent chain of the given order and chain-degree; the paper supplies such chains only for the specific iterated-exponential format with $n$35. The combined $n$36-$n$37 construction doubles the chain order, so simultaneous sharpness at order $n$38 remains open. Most significantly, the dependence on $n$39 is not resolved: the $n$40 factor may be improvable, but no construction attains it beyond the trivial polynomial case $n$41, and the $n$42 lower bound rules out only improvements growing slower than $n$43. Whether the true optimum lies strictly between $n$44 and $n$45 is unresolved.

Conclusion

This paper converts Khovanskii's bound from an unexamined upper estimate into a partially calibrated one: its growth in the chain-degree $n$46 (as $n$47) and in the degrees $n$48 (as $n$49) is asymptotically optimal, while its $n$50 dependence on the order appears loose, with concrete small-parameter gaps and known improvements in the fewnomial and $n$51 settings. The methods — inverse-branch dynamics for the $n$52 construction and evaluation-map dimension counting for the $n$53 construction — are elementary but effective, and the sharpness results delimit precisely what any future strengthening of the bound could achieve.