Correction To: Found Comput Math

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 \(q\le 2^{\underline{L}(f)^2}\).


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

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

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 [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 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 \(\underline{L}^{ ns } = \underline{L}_\infty ^{ ns } = \underline{L}_q^{ ns }\).