## Correction To: Found Comput Math https://doi.org/10.1007/s10208-002-0059-5

Vladimir Lysikov kindly pointed out an error in the proof of Theorem 5.7. We provide here a corrected statement and its proof.

### Theorem 5.7

For polynomials f over an algebraically closed field k, we have $$\underline{L}_{q}(f) \le 2 \underline{L}(f)$$ with $$q\le 2^{\underline{L}(f)^2}$$.

### Proof

We proceed as in Lehmkuhl and Lickteig , who proved a similar bound on the order of approximation for border rank (approximative bilinear complexity).

The proof is based on the following geometric description of the set $$\{f\in A_n \mid L(f) \le r\}$$. The field k is assumed to be algebraically closed. A straight-line program $$\Gamma$$ is a description for a computation of a polynomial from constants $$z_1,\ldots ,z_m$$ and variables $$X_1,\ldots ,X_n$$ (recall that we do not allow divisions). Let $$\phi _\Gamma (z)$$ denote the polynomial in $$A_n:=k[X_1,\ldots ,X_n]$$ computed by $$\Gamma$$ from the list of constants $$z\in k^m$$. Let $$r_*$$ denote the number of multiplication instructions of $$\Gamma$$. Then, we have

\begin{aligned} \phi _\Gamma (z) = \sum _\mu \phi _{\Gamma ,\mu }(z) X^\mu , \end{aligned}

where the sum runs over all $$\mu \in \mathbb {N}^n$$ with $$\mu _1+\ldots + \mu _n \le 2^{r_*}$$. Moreover, the coefficient polynomials $$\phi _{\Gamma ,\mu }(z)$$ have degree at most $$2^{r_*}$$. We interpret $$\phi _{\Gamma }$$ as a morphism $$k^m \rightarrow \{f\in A_n \mid \deg f \le 2^{r_*}\}$$ of affine varieties. Applying [1, Theorem 8.48] to the polynomial map $$z\mapsto (z,\phi _\Gamma (z))$$, we see that $$\deg \mathrm {graph}(\phi _\Gamma ) \le \big (2^{r_*}\big )^m =: D$$. The image $$\mathcal{C}_\Gamma$$ of $$\phi _\Gamma$$ is an irreducible, constructible set. We have for fixed r that

\begin{aligned} \{f\in A_n \mid L(f) \le r\} = \bigcup _{\Gamma } \mathcal{C}_\Gamma , \end{aligned}

where the union is over all straight-line programs $$\Gamma$$ of length r.

Assume now that f is in the Zariski-closure of the set on the left-hand side. Then, we have $$f\in \overline{\mathcal{C}_\Gamma }$$ for some $$\Gamma$$. (We remark that in the case $$k=\mathbb {C}$$ the Zariski-closure of constructible sets coincides with the closure with respect to the Euclidean topology (cf. [3, Theorem 2.33]).

We apply now two results proven in Lehmkuhl and Lickteig  to the morphism $$\phi _\Gamma$$. Proposition 1 of  claims that there exists an irreducible curve $$C\subseteq k^m$$ such that $$f\in \overline{\phi _\Gamma (C)}$$ and $$\deg C \le \deg \mathrm {graph}(\phi _\Gamma )$$. The Corollary to Proposition 3 in  states that there exists a point $$\zeta =(\zeta _1,\ldots ,\zeta _m)\in k((\epsilon ))^m$$ such that $$F:=\phi _\Gamma (\zeta )$$ is defined over $$k[[\epsilon ]]$$, satisfies $$F_{\epsilon =0} =f$$ and such that all formal Laurent series $$\zeta _i$$ have order at least $$-\deg C$$. We conclude with Lemma 5.6(2) that $$L(F)\le r$$ and hence $$\underline{L}(f)\le r$$, which proves the nontrivial direction of Theorem 2.4. Moreover, we have shown that there is a straight-line program of length r, which computes F in $$k((\epsilon ))[X]$$ from the X-variables and constants $$\zeta _i$$ having order at least $$-\deg C \ge - D$$. By a similar reasoning as in the proof of Lemma 5.6(1), we can construct from this a straight-line program of length at most 2r, which computes in $$k[[\epsilon ]][X]$$ an element of the form $$\epsilon ^{q} f + \epsilon ^{q+1} f'$$ with $$q \le 2^{r_*} D = 2^{(m+1)r_*}$$. We therefore have $$\underline{L}_q(f) \le 2 r$$. To complete the proof, we note that $$(m+1)r_*\le r^2$$, unless $$m=r$$ and $$r=r_*$$. However, in this case, the components of $$\phi _\Gamma$$ have degree at most 1 and we get $$q \le 2^{r_*}$$ since $$\deg \mathrm {graph}\phi _\Gamma \le 1$$. $$\square$$

By tracing the proofs of the above results, it is straightforward to show the following statement.

### Remark 5.8

By counting only nonscalar multiplications, one can introduce the notions $$\underline{L}^{ ns }, \underline{L}_q^{ ns }$$ in an analogous way. We then have $$\underline{L}^{ ns } = \underline{L}_\infty ^{ ns } = \underline{L}_q^{ ns }$$.