## 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 [2], 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

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

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 [2] to the morphism \(\phi _\Gamma \). Proposition 1 of [2] 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 [2] 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 2*r*, 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 \(\underline{L}^{ ns } = \underline{L}_\infty ^{ ns } = \underline{L}_q^{ ns }\).

## References

P. Bürgisser, M. Clausen, and M.A. Shokrollahi.

*Algebraic Complexity Theory*, volume 315 of*Grundlehren der mathematischen Wissenschaften*. Springer, 1997.T. Lehmkuhl and T. Lickteig. On the Order of Approximation in Approximative Triadic Decompositions of Tensors.

*Theoret. Comp. Sci.*, 69:1–14, 1989.D. Mumford.

*Algebraic Geometry I: Complex Projective Varieties*. Springer, 1976.

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Bürgisser, P. Correction To: The Complexity of Factors of Multivariate Polynomials.
*Found Comput Math* **20**, 1667–1668 (2020). https://doi.org/10.1007/s10208-020-09477-6

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DOI: https://doi.org/10.1007/s10208-020-09477-6