# One-sided root bounds and Taylor order

*Written by GPT-6.1 Sol (OpenAI), October 2026.  Public domain (CC0).*

A root can approach zero much more slowly than its parameter. The equation \(z^2=\lambda\) gives the simplest example: the root size is \(|\lambda|^{1/2}\). A linear bound on upper imaginary parts rules out such fractional scales when it holds for both signs of a real parameter.

Read [Cauchy bounds, root counts and analytic extensions][cauchy] for Taylor expansions, multiplicities and Rouché's theorem. Basic references are that lesson, Puiseux's *Recherches sur les fonctions algébriques*, and Helmke's *On the Newton-Puiseux Series*. We use only a finite lowest-weight polynomial and the proved root-count theorem.

## 1. Axis order and total order measure different things

Let \(g\) be holomorphic near \((0,0)\in\mathbb C^2\), with expansion

\[
g(z,\lambda)=\sum_{j,\ell\ge0}a_{j\ell}z^j\lambda^\ell.
\tag{11}
\]

Suppose \(g(z,0)\) has exact zero order \(d\). This means \(a_{d0}\ne0\) and \(a_{j0}=0\) for \(j<d\). The total Taylor order instead minimizes \(j+\ell\) over all nonzero coefficients. It is at most \(d\). For \(z^3-\lambda^2\), the axis order is three and the total order is two.

**The root-bound theorem.** If every nearby zero with a real parameter satisfies

\[
\operatorname{Im}z\le C|\lambda|
\tag{12}
\]

for one fixed constant \(C\ge0\) and for both positive and negative parameters, then the total Taylor order is \(d\). Thus every derivative \(\partial_z^j\partial_\lambda^\ell g(0,0)\) with \(j+\ell<d\) vanishes.

The full proof is [Theorem 4.1](one-sided-root-bounds-and-taylor-order-formal.md). The coefficients may be complex and the roots may be repeated. Only an upper bound is assumed. The case \(d=0\) has a nonzero constant term; the case \(d=1\) follows already from \(g(0,0)=0\) and \(a_{10}\ne0\).

## 2. A fractional scale exposes a forbidden root cluster

Suppose a term of total degree less than \(d\) were present. For each \(j<d\), take the first nonzero parameter exponent \(\ell_j\), if one exists. Put

\[
\alpha=\min_j\frac{\ell_j}{d-j}.
\tag{13}
\]

This is a positive rational number less than one. The substitution \(z=r^\alpha u,\ \lambda=\sigma r\), where \(r>0\) and \(\sigma=1\) or \(-1\), gives

\[
r^{-\alpha d}g(r^\alpha u,\sigma r)
\longrightarrow
H_\sigma(u):=
\sum_{\alpha j+\ell=\alpha d}a_{j\ell}\sigma^\ell u^j.
\tag{14}
\]

The convergence is uniform on every fixed compact disk. Every omitted term has positive excess weight; absolute convergence of the original series controls all omitted terms together. The leading coefficient of \(H_\sigma\) is still \(a_{d0}\).

For the polynomial
\(h=z^6-2\lambda z^3+\lambda^2+z^7+2\lambda^3\),
the weight is \(\alpha=1/3\). The first three terms have weight two, while the last two have larger weights. Consequently \(H_+(u)=(u^3-1)^2\).

![The exponent pairs of five nonzero monomials lie on or above the line j/3 plus ell equals two.](figures/lowest-weight-edge.png)

*Figure 1. The horizontal coordinate is the exponent \(j\) of \(z\), and the vertical coordinate is the exponent \(\ell\) of \(\lambda\). The three orange points \((6,0),(3,1),(0,2)\) lie on \(j/3+\ell=2\) and produce \(H_+=(u^3-1)^2\). The black points \((7,0),(0,3)\) produce terms that vanish after scaling. The dashed line \(j+\ell=2\) records the total order, which is two rather than the axis order six. This is the complete finite monomial support of the displayed polynomial. See complete-proof Lemmas 2.1–2.2 and Solution 3.*

Write \(\alpha=p/q\) in lowest terms. The equality \(p(d-j)=q\ell\) forces all exponents in \(H_+\) to be congruent modulo \(q\). Its nonzero roots therefore come in orbits under rotation by \(2\pi/q\).

If \(q\ge3\), any nonzero orbit has a root in the upper half-plane. Its points sum to zero and cannot all be real. If \(q=2\), a real orbit might consist of two real points. Then \(\alpha=1/2\), and changing the parameter sign rotates those roots by \(i\).

![Changing the parameter sign rotates two real roots into imaginary roots; a three-point root orbit already has an upper root.](figures/root-orbits-and-the-parameter-sign.png)

*Figure 2. The left panel uses \(z^2-\lambda\): the limiting roots for \(\lambda>0\) are \(u=\pm1\), while for \(\lambda<0\) they are \(u=\pm i\). The arrows show multiplication by \(i\). The right panel uses \(H_+=(u^3-1)^2\), whose distinct roots are \(1,(-1+i\sqrt3)/2,(-1-i\sqrt3)/2\), each of multiplicity two. The orange root has positive imaginary part. These are exact limiting-polynomial roots in the \(u\)-plane; roots of the perturbed polynomial are obtained through Rouché's theorem. See complete-proof Lemma 3.1 and Solutions 2–4.*

Choose a small circle entirely in the upper half-plane around such a root of \(H_\sigma\). Rouché retains its full positive root count for small \(r\). Each retained root has

\[
\operatorname{Im}z_r\ge c r^\alpha
\quad\text{for some }c>0.
\tag{15}
\]

Since \(\alpha<1\), the ratio \(r^\alpha/r=r^{\alpha-1}\) tends to infinity. This contradicts (12). A repeated limiting root works equally well: Rouché counts its multiplicity without requiring it to remain a single repeated root.

## 3. Four examples

### Example 1: a complex repeated root satisfies the bound sharply.

Let \(a=2+3i\) and

\[
g(z,\lambda)=e^{z-\lambda}(z-a\lambda)^4.
\tag{16}
\]

The exponential is nonzero, so the only root is \(z=a\lambda\), of multiplicity four. For real \(\lambda\), its imaginary part is \(3\lambda\le3|\lambda|\). Both parameter signs are allowed, although the coefficients are not real. The axis order is four. The first homogeneous part is \((z-a\lambda)^4\), of total degree four, so the total order is also four. Every derivative of total order less than four vanishes.

### Example 2: one parameter sign is insufficient.

For \(g=z^2-\lambda\), nonnegative real \(\lambda\) gives the two real roots \(\pm\sqrt\lambda\); thus an upper bound with \(C=0\) holds on that half-line. For \(\lambda=-r<0\), the roots are \(\pm i\sqrt r\). The upper root has \(\operatorname{Im}z=\sqrt r\), exceeding \(Cr\) for every fixed \(C\) and sufficiently small \(r\). The axis order two and total order one therefore do not contradict the theorem: its two-sign hypothesis fails.

### Example 3: a nonpolynomial germ has a repeated limiting root.

Take

\[
g(z,\lambda)=e^{z+2\lambda}(z^3-\lambda)^2.
\tag{17}
\]

Its axis order is six and total order is two. With \(\lambda=r>0\), the three distinct roots are \(r^{1/3}\) times the cube roots of one, each repeated twice. In particular the upper root has imaginary part \((\sqrt3/2)r^{1/3}\), which violates any linear bound. The scaled functions converge to \((u^3-1)^2\); the nonzero exponential contributes a factor tending uniformly to one on fixed disks. The full analytic series need not terminate.

### Example 4: the numerator of the fractional weight need not be one.

For \(g=z^3-\lambda^2\), the axis order is three, total order two, and \(\alpha=2/3\). Both parameter signs give the same three roots, with maximum imaginary part
\((\sqrt3/2)|\lambda|^{2/3}\).
The root bound with exponent \(2/3\) holds exactly, but the linear bound does not. The denominator of the reduced weight controls the rotational orbit; the numerator controls the scale.

## 4. Exercises

1. **Basic, 10 points.** For the germ in Example 1, find its first homogeneous part and the coefficients of \(z^3\lambda\) and \(z^2\lambda^2\). Determine the smallest constant in (12) for real parameters of both signs, and explain why the exponential changes neither the zeros nor the two orders.
2. **Basic, 12 points.** For \(z^2-\lambda\), verify the bound for nonnegative real parameters and its failure for negative ones. Repeat for \((z^2-\lambda)^k\), \(k\ge1\), including both orders and every root multiplicity.
3. **Intermediate, 12 points.** For \(h=z^6-2\lambda z^3+\lambda^2+z^7+2\lambda^3\), compute the weight (13), both limiting polynomials and the exact scaled error terms. Explain every marked point and both lines in Figure 1.
4. **Intermediate, 12 points.** Prove that an orbit \(\beta e^{2\pi i j/q}\), \(0\le j<q\), with \(\beta\ne0,\ q\ge3\), contains a root with positive imaginary part. Explain the obstruction at \(q=2\), and prove the parameter-sign identity (8) of the complete proof without assuming real coefficients.
5. **Intermediate, 14 points.** Prove uniform convergence of both scaled functions for Example 3 on \(|u|\le B\). Then give the summable majorant which justifies (14) for an arbitrary convergent germ, including its infinitely many higher-weight terms.
6. **Intermediate, 12 points.** Replace (12) by \(\operatorname{Im}z\le C|\lambda|^\gamma\), with \(\gamma>0\). For \(z^3-\lambda^2\), determine exactly which \(\gamma\) permit such a bound near zero, for both real parameter signs. Include a valid constant and a sequence disproving every excluded exponent.
7. **Advanced, 14 points.** Let \(g(z,w)\) be holomorphic near zero with \(w\in\mathbb C^k\), and let \(g(z,0)\) have axis order \(d\). Suppose every local zero with \(w\in\mathbb R^k\) satisfies \(\operatorname{Im}z\le C|w|\). Use the one-parameter theorem to prove total order \(d\). Check the sharp complex example \((z-i(a+2b))^3\) and the failure for \(z^3-a^2-b^2\).
8. **Advanced, 14 points.** Let \(\omega=(-1+i\sqrt3)/2\) and let \(h\) be the polynomial in Exercise 3. For \(0<r\le10^{-6}\), prove that \(r^{-2}h(r^{1/3}u,r)\) has exactly two zeros, counted with multiplicity, in \(|u-\omega|<1/10\). Use exact boundary estimates, deduce \(\operatorname{Im}z>(3/4)r^{1/3}\) for these roots, and show why this violates any proposed linear bound.

## 5. Complete solutions

### Solution 1

The exponential has constant term one. Its remaining terms increase total degree, so the first homogeneous part is
\[
(z-a\lambda)^4
=z^4-4az^3\lambda+6a^2z^2\lambda^2
-4a^3z\lambda^3+a^4\lambda^4.
\]
The requested coefficients are \(-4a=-8-12i\) and \(6a^2=-30+72i\), since \(a^2=-5+12i\). No lower-total-degree term exists. At \(\lambda=0\), the germ is \(e^z z^4\), so the axis order is four as well.

The exponential never vanishes, with reciprocal \(e^{-z+\lambda}\). It leaves the root \(a\lambda\) and its fourth multiplicity unchanged. Its imaginary part equals \(3\lambda\), so \(C=3\) works for both signs. For every small positive \(\lambda\), equality forces \(C\ge3\). Thus three is the smallest possible constant.

### Solution 2

For \(\lambda\ge0\), the roots \(\pm\sqrt\lambda\) are real, including the double root at zero. For \(\lambda=-r<0\), they are \(\pm i\sqrt r\); the positive imaginary part divided by \(|\lambda|=r\) is \(r^{-1/2}\), which tends to infinity.

Raising the polynomial to the \(k\)-th power changes each nonzero-parameter root multiplicity to \(k\). At \(\lambda=0\) there is one root of multiplicity \(2k\). Its axis order is \(2k\). In the binomial expansion the term \((-\lambda)^k\) has total degree \(k\), and all other terms have degree \(k+j\), where \(j\) counts the chosen \(z^2\)-factors. Thus the total order is \(k\). The same root locations prove the same one-sign bound and the same failure on the other sign, for every \(k\).

### Solution 3

The nonzero support pairs \((j,\ell)\) are \((6,0),(3,1),(0,2),(7,0),(0,3)\), with coefficients \(1,-2,1,1,2\). The first parameter exponents for \(j<6\) are \(\ell_3=1,\ell_0=2\). Both ratios in (13) are \(1/3\). The additional \(\ell=3\) term at \(j=0\) does not alter the minimum. The axis order is six and the total order is two.

Direct substitution gives
\[
r^{-2}h(r^{1/3}u,\sigma r)
=u^6-2\sigma u^3+1+r^{1/3}u^7+2\sigma r.
\tag{18}
\]
Hence \(H_+=(u^3-1)^2\) and \(H_-=(u^3+1)^2\). The two displayed error terms vanish uniformly on a fixed disk, since their absolute values are at most \(r^{1/3}B^7+2r\).

The three edge pairs have \(j/3+\ell=2\). The other two have weights \(7/3\) and three, respectively. Their locations explain the solid line and which terms survive. The dashed line has \(j+\ell=2\) and meets the support at \((0,2)\); all other support pairs have strictly larger total degree. It records the total order, not the fractional weight.

### Solution 4

Put \(\eta=e^{2\pi i/q}\). The \(q\) orbit points are distinct, have one nonzero modulus, and sum to
\(\beta(1+\eta+\cdots+\eta^{q-1})=0\), because \(\eta\ne1\) and \(\eta^q=1\). If none had positive imaginary part, all would lie on or below the real axis. At least one must be nonreal, since a fixed nonzero modulus has only two real points and \(q\ge3\). The sum of imaginary parts would then be strictly negative, contradicting the zero sum.

At \(q=2\), the orbit \(\{1,-1\}\) is entirely real and its imaginary parts sum to zero. A reduced weight in \((0,1)\) with denominator two is \(1/2\). For every edge pair,
\((-1)^\ell e^{i\pi\alpha j}=e^{i\pi(\ell+\alpha j)}=e^{i\pi\alpha d}\).
Summing its coefficient times \(u^j\) proves
\(H_-(e^{i\pi\alpha}u)=e^{i\pi\alpha d}H_+(u)\).
This computation holds for each arbitrary complex coefficient. A positive real root of \(H_+\) therefore becomes a positive imaginary root of \(H_-\).

### Solution 5

For either sign, Example 3 gives exactly
\[
r^{-2}g(r^{1/3}u,\sigma r)
=e^{r^{1/3}u+2\sigma r}(u^3-\sigma)^2.
\]
On \(|u|\le B\), the exponent has modulus at most \(r^{1/3}B+2r\). The exponential series gives
\(|e^v-1|\le |v|e^{|v|}\), so the exponential tends uniformly to one. The other factor is bounded by \((B^3+1)^2\). The limits are \((u^3-1)^2\) and \((u^3+1)^2\), respectively.

For a general germ, choose \(r_0>0\) such that
\(\sum_{j,\ell}|a_{j\ell}|(B r_0^\alpha)^j r_0^\ell<\infty\).
Every scaled term is bounded by this summand times \(r_0^{-\alpha d}\), because its additional factor
\((r/r_0)^{\alpha j+\ell-\alpha d}\) is at most one. Take a tail with arbitrarily small total, uniformly for \(0<r\le r_0\). The remaining finite sum converges uniformly to its zero-excess-weight terms. This proves the uniform limit with the entire infinite tail included; bounding finitely many selected terms alone would be insufficient.

### Solution 6

For \(|\lambda|=r>0\), both signs give the roots \(r^{2/3}\), \(r^{2/3}\omega\) and \(r^{2/3}\overline\omega\). Their maximum imaginary part is \((\sqrt3/2)r^{2/3}\).

For \(0<\gamma\le2/3\) and \(0<r\le1\), \(r^{2/3}\le r^\gamma\), so \(C=\sqrt3/2\) works, including the zero parameter. If \(\gamma>2/3\), the ratio of the upper imaginary part to \(r^\gamma\) is \((\sqrt3/2)r^{2/3-\gamma}\), which tends to infinity. Taking positive \(\lambda_n=1/n\) supplies a sequence disproving every finite constant. Thus exactly \(0<\gamma\le2/3\) work. The linear exponent is excluded.

### Solution 7

The case \(k=0\) has only the axis and therefore total order \(d\). Assume \(k\ge1\). For every real vector \(v\), the restricted germ \(g_v(z,\lambda)=g(z,\lambda v)\) has the same axis order \(d\). Every local zero with real \(\lambda\) obeys
\(\operatorname{Im}z\le C|v||\lambda|\). The one-parameter theorem makes every homogeneous Taylor part of degree less than \(d\) vanish after this restriction.

If such a part \(P_N(z,w)\) of \(g\) were nonzero, write
\(P_N(z,w)=\sum_{j=0}^N z^j p_{N-j}(w)\), with each coefficient homogeneous in \(w\). Its substitution is
\(\sum_j z^j\lambda^{N-j}p_{N-j}(v)\).
Vanishing for every real \(v\) forces every \(p_{N-j}\) to vanish on \(\mathbb R^k\). A complex polynomial vanishing on all real vectors is identically zero: in one coordinate it has infinitely many real zeros, so each coefficient in that coordinate vanishes; applying the same argument successively to the remaining coordinates proves the claim. Thus \(P_N=0\), a contradiction. The nonzero axis coefficient gives total order exactly \(d\).

For \((z-i(a+2b))^3\), the sole root has multiplicity three and imaginary part \(a+2b\le\sqrt5\sqrt{a^2+b^2}\) by Cauchy–Schwarz. The polynomial is homogeneous of total degree three and its axis is \(z^3\). For \(z^3-a^2-b^2\), the total order is two while the axis order is three. Restricting to \(b=0,\ a=r>0\) gives the upper imaginary part \((\sqrt3/2)r^{2/3}\), contradicting any bound \(Cr\).

### Solution 8

The disk centered at \(\omega\) with radius \(1/10\) contains only the double root \(\omega\) of \(H_+=(u^3-1)^2\). The distances from \(\omega\) to the other two cube roots are both \(\sqrt3\). On its boundary,
\[
|u^3-1|
=|u-\omega||u-1||u-\overline\omega|
>\frac1{10}\left(\frac85\right)^2,
\]
because \(\sqrt3>17/10\). Consequently
\(|H_+|>4096/62500\).

Also \(|u|\le11/10\). Equation (18) bounds the perturbation for \(r\le10^{-6}\) by
\[
\frac1{100}\left(\frac{11}{10}\right)^7
+\frac2{10^6}
=\frac{19489171}{10^9}
<\frac{4096}{62500}.
\tag{19}
\]
[Rouché's theorem][rouche] therefore retains exactly two zeros, counted with multiplicity, in the disk. Its points have
\(\operatorname{Im}u>\sqrt3/2-1/10>3/4\).
Both corresponding zeros of \(h(z,r)\) satisfy
\(\operatorname{Im}z>(3/4)r^{1/3}\); their real and imaginary coordinates tend to zero because the disk is bounded.

For \(C>0\), choose \(r>0\) smaller than both \(10^{-6}\) and \((3/(4C))^{3/2}\), and small enough for any proposed product neighborhood. Then \((3/4)r^{1/3}>Cr\). For \(C=0\), every such upper-half-plane zero already violates the bound. The conclusion concerns two zeros in the displayed small disk and does not require a global count of all zeros of the degree-seven polynomial.

## References

- [Cauchy bounds, root counts and analytic extensions][cauchy], Lemma 1.1 and Corollary 2.3, for the expansion and persistent root clusters. Its linked components retain their stated terms.
- Victor A. Puiseux, *Recherches sur les fonctions algébriques*, Journal de Mathématiques Pures et Appliquées, series 1, volume 15 (1850), 365–480. [Original article](https://www.numdam.org/item/JMPA_1850_1_15__365_0/).
- Stefan Helmke, *On the Newton-Puiseux Series*, Kyoto University Research Institute for Mathematical Sciences, 2018. [Lecture description and historical references](https://www.kurims.kyoto-u.ac.jp/kenkyubu/zengaku/18/helmke.html).

[cauchy]: ../../AN02-L045.html#cauchy-estimates-and-parameter-contours
[rouche]: ../../AN02-L045.html#modulus-and-multiplicity-on-a-disk