Proof Let $X = \{u \in U \colon \rho _1(u,u) = \dots = \rho _k(u,u) = 0\}$ . Note that

so we need to estimate

for nonzero $\lambda $ . Write $\phi = \lambda _1 \rho _1 + \dots + \lambda _k \rho _k$ . Then

where we used the Gowers–Cauchy–Schwarz inequality in the last step. Taking 4th power and expanding, we get

However, it is a standard fact that the last expression equals $2^{-\operatorname {rank}(\phi + \phi \circ (1\,2))}$ (e.g., by Lemma 7 in [Reference Milićević28]). From assumption on forms $\rho _1, \dots , \rho _k$ , we see that

. The lemma now follows.

Proof Let $G = U \oplus \langle w_1, \dots , w_d\rangle $ . Then there exist linear maps $\pi \colon G \to U$ and $\phi _1, \dots , \phi _d \colon G \to \mathbb {F}_2$ such that $x = \pi (x) + \sum _{i \in [d]} \phi _i(x) w_i$ for all $x \in G$ . Then

$$ \begin{align*}\alpha(x_1, \dots, x_k) = &\ \alpha(\pi(x_1) + \phi_1(x_1) w_1 + \dots + \phi_d(x_1) w_d, x_2, \dots, x_k) \\ = &\sum_{i \in [d]} \phi_i(x_1) \alpha(w_i, x_2, \dots, x_k) + \alpha(\pi(x), x_2, \dots, x_k)\\ = & \cdots\\ = & \left(\sum_{c \in [k]} \sum_{i \in [d]} \phi_i(x_c) \alpha(\pi(x_1), \dots, \pi(x_{c-1}), w_i, x_{c+1}, \dots, x_k)\right)\\ & \quad + \alpha(\pi(x_1), \dots, \pi(x_k)).\\[-34pt] \end{align*} $$

Proof Let the maps $\pi , \phi _1, \dots , \phi _d$ be as in the proof of the previous lemma. Define the multilinear form $\tilde {\sigma } \colon G^k \to \mathbb {F}_2$ by $\tilde {\sigma }(x_1, \dots , x_k) = \sigma (\pi (x_1), \dots , \pi (x_k))$ . Since $\sigma $ is symmetric on $U \times \cdots \times U$ , we have that $\tilde {\sigma }$ is symmetric. On the other hand, since $\pi $ is a projection onto U, we see that $\tilde {\sigma }|_{U \times \cdots \times U} = \sigma $ and hence

$$\begin{align*}\operatorname{prank}\Big((\alpha + \tilde{\sigma})|_{U \times\cdots\times U}\Big) \leq r.\end{align*}$$

By the previous lemma, we conclude that $\operatorname {prank}(\alpha + \tilde {\sigma }) \leq kd + r$ , as required.

Proof Using the notion of the large multilinear spectrum (Definition 8 in [Reference Milićević27]), and writing $c' = c^{2^{-k}}$ , we see that the condition is that $\alpha \in \operatorname {Spec}^{\text {ml}}_{c'} f$ . We need to show that $\beta \in \operatorname {Spec}^{\text {ml}}_{c"} f$ for some $c" \geq 2^{-2r} c'$ . This follows from the proof of Lemma 39 in [Reference Milićević27], which has a large bias assumption instead of a low partition rank assumption.

Proof Let W be a subspace such that $G = U \oplus W$ . Using this decomposition, we get

By the triangle inequality, we get

By averaging, there exists a choice of $y, b_1, \dots , b_k \in W$ such that

We introduce additional variables $a^{\prime }_1, \dots , a^{\prime }_k \in U$ and make a change of variables where we replace x by $x + a_1 + \dots + a_k$ and $a_i$ by $a_i + a^{\prime }_i$ to obtain

By the triangle inequality,

For each $x \in U$ , we may define auxiliary functions $f_{I, x} \colon G^k \to \mathbb {D}$ for $I \subseteq [k]$ , each of the form

$$\begin{align*}f_{I, x}(z_1, \dots, z_k) = f\Bigg(x +y + \sum_{i \in I} b_i + z_1 + \dots + z_k\Bigg) (-1)^{\alpha(z_1, \dots, z_k)} \ell_I(z_1, \dots, z_k),\end{align*}$$

where $\ell _I$ is a product of several factors, each depending on a proper subset of variables $z_1, \dots , z_k$ , coming from terms that arise from the expansion of

$$\begin{align*}(-1)^{\alpha(a_1 + a^{\prime}_1 + b_1, \dots, a_k + a^{\prime}_k + b_k) + \alpha(a_1 + a^{\prime}_1, \dots, a_k + a^{\prime}_k)}\end{align*}$$

and whose values are of modulus 1. In the new notation, we get

Using the Gowers–Cauchy–Schwarz inequality, we get

and Hőlder’s inequality implies that

Thus, there is a choice of subset $I \subseteq [k]$ such that $c \leq \mathop {\mathbb {E}}_x \|f_{I, x}\|_{\square ^k}^{2^k}$ . By definition, writing $\tilde {b} = \sum _{i \in I} b_i$ and splitting $\ell _I(z_1, \dots , z_k)$ into further factors that do not depend on a variable in $z_{[k]}$ , we have

$$\begin{align*}f_{I, x}(y_1, \dots, y_k) = f(x + y_1 + \dots + y_k + y + \tilde{b}) (-1)^{\alpha(y_1, \dots, y_k)} \prod_{i \in [k]} \ell_i(y_{[k] \setminus \{i\}})\end{align*}$$

for some functions $\ell _i$ with $|\ell _i(y_{[k] \setminus \{i\}})| = 1$ for all $y_{[k] \setminus \{i\}}$ . Let $\tilde {f} \colon U \to \mathbb {D}$ be given by $\tilde {f}(x) = f(x + y + \tilde {b})$ .

Thus,

Proof Let $n = \dim G$ and fix a basis of G. Write $\beta $ as a linear combination of monomials. Thus,

$$\begin{align*}\sigma(x_1, x_2, \dots, x_k) = \sum_{i_1, \dots, i_k \in [n]} \lambda_{i_1, \dots, i_k} x_{1\,i_1} \dots x_{k\,i_k}.\end{align*}$$

Define coefficients $\mu _{j_1, \dots , j_{k+1}}$ as follows: if $j_1, \dots , j_{k+1}$ are all distinct, then put $\mu _{j_1, \dots , j_{k+1}} = 0$ , and in the other case, set $\mu _{j_1, \dots , j_{k+1}} = \lambda _{i_1, \dots , i_k}$ for any sequence $i_1, \dots , i_k$ with the property that there is some element $i_\ell $ which, if repeated, gives a permutation of the sequence $j_1, \dots , j_{k+1}$ . We need to check that this is well defined. Suppose that $i^{\prime }_1, \dots , i^{\prime }_k$ is another such sequence, and that $i^{\prime }_m$ is the element that is repeated. If $i^{\prime }_m = i_\ell $ , then i and $i'$ are the same up to reordering, so by symmetry of $\sigma $ we have the equality $\lambda _{i_1, \dots , i_k} = \lambda _{i^{\prime }_1, \dots , i^{\prime }_k}$ , as desired. On the other hand, if $i_\ell \not = i^{\prime }_m$ , then that means that the number of times the element $i^{\prime }_m$ appears in the sequence i is one greater than the number of times it appears in the sequence $i'$ , and, similarly, $i_\ell $ has one occurrence more in $i'$ than it has in i, while other elements appear an equal number of times in both sequences. Since $\sigma $ is strongly symmetric, we again have equality $\lambda _{i_1, \dots , i_k} = \lambda _{i^{\prime }_1, \dots , i^{\prime }_k}$ . Coefficients $\mu $ are symmetric directly from the definition, so defining $\tilde {\sigma }$ as

$$\begin{align*}\tilde{\sigma}(x_0, x_1, x_2, \dots, x_k) = \sum_{i_0, i_1, \dots, i_k \in [n]} \mu_{i_0, i_1, \dots, i_k} x_{0\,i_0} x_{1\,i_1} \dots x_{k\,i_k}\end{align*}$$

produces a symmetric multilinear form. Finally,

$$\begin{align*}&\tilde{\sigma}(x_1, x_1, x_2, \dots, x_k) = \sum_{i_1, \dots, i_k \in [n]} \mu_{i_1, i_1, \dots, i_k} x_{1\,i_1} \dots x_{k\,i_k} = \sum_{i_1, \dots, i_k \in [n]} \lambda_{i_1, i_2, \dots, i_k} x_{1\,i_1} \dots x_{k\,i_k} \\&\quad= \sigma(x_1, x_2, \dots, x_k).\end{align*}$$

This means that $\tilde {\sigma }(x_1, x_1, x_2, \dots , x_k)$ is symmetric, and thus $\tilde {\sigma }$ is strongly symmetric.

Proof of Theorem 1.6

During the proof, we consider decompositions of the shape

(4.6) $$ \begin{align}&\alpha(x_1, x_2, x_3, \dots, x_k) + \alpha(x_2, x_1, x_3, \dots, x_k) = \sum_{i \in [r_1]} \beta_i(x_1, x_2, x_{I_{i}}) \beta^{\prime}_{i, 1}(z_{I^{\prime}_{i, 1}}) \cdots \beta^{\prime}_{i, d_i}(z_{I^{\prime}_{i, d_i}})\nonumber \\&\qquad+ \sum_{i \in [r_2]} \gamma_i(x_1, x_{J_{i, 1}}) \gamma^{\prime}_i(x_2, x_{J_{i, 2}})\gamma^{\prime\prime}_{i, 1}(x_{J^{\prime}_{i, 1}}) \cdots \gamma^{\prime\prime}_{i, d^{\prime}_i}(x_{J^{\prime}_{i, d^{\prime}_i}})\end{align} $$

for some integers $r_1$ and $r_2$ and suitable multilinear forms where the partitions of variables in products belong to some downset of partitions $\mathcal {P}$ . We write $r' = r_1 + r_2$ , which will satisfy $r' \leq O(\exp ^{(O(1))}(r))$ at all times. Initially, we have $r' \leq r$ and $\mathcal {P}$ contains all partitions except the trivial one $\{[k]\}$ .

The proof splits into three parts: in the first stage, we remove the $\beta _i(x_1, x_2, x_{I_{i}}) \beta ^{\prime }_{i, 1}(z_{I^{\prime }_{i, 1}}) \cdots \beta ^{\prime }_{i, d_i}(z_{I^{\prime }_{i, d_i}})$ terms, in the second, we remove the almost all of $ \gamma _i(x_1, x_{J_{i, 1}}) \gamma ^{\prime }_i(x_2, x_{J_{i, 2}})\gamma ^{\prime \prime }_{i, 1}(x_{J^{\prime }_{i, 1}}) \cdots \gamma ^{\prime \prime }_{i, d^{\prime }_i}(x_{J^{\prime }_{i, d^{\prime }_i}})$ terms, and in the final step, we remove remaining terms which will have linear forms in $x_1$ and $x_2$ , thereby relating $\alpha $ to a form symmetric in the first two variables.

Step 1. Removing $\beta $ forms. Using Proposition 3.3, we may assume that every $\beta _i$ comes from a slice of $\alpha + \alpha \circ (1\,\,2)$ . This has the cost of replacing r by $r' = O({r}^{O(1)})$ . Misusing the notation, we still write $r_1$ and $r_2$ for the number of products in each of the two sums. Thus, for each $i \in [r_1]$ , we obtain a multilinear form $\tilde {\beta }_i(x_1, x_2, x_{I_{i}})$ such that $\beta _i(x_1, x_2, x_{I_{i}}) = \tilde {\beta }_i(x_1, x_2, x_{I_{i}}) + \tilde {\beta }_i(x_2, x_1, x_{I_{i}})$ . Considering the multilinear form $\alpha '$ defined by

$$\begin{align*}\alpha'(x_{[k]}) = \alpha(x_{[k]}) + \sum_{i \in [r_1]} \tilde{\beta}_i(x_1, x_2, x_{I_{i}}) \beta^{\prime}_{i, 1}(z_{I^{\prime}_{i, 1}}) \cdots \beta^{\prime}_{i, d_i}(z_{I^{\prime}_{i, d_i}}),\end{align*}$$

we see that $\operatorname {prank}^{\mathcal {P}} (\alpha + \alpha ') \leq O({r}^{O(1)})$ and

(4.7) $$ \begin{align}&\alpha'(x_1, x_2, x_3, \dots, x_k) + \alpha'(x_2, x_1, x_3, \dots, x_k) \\&\quad= \sum_{i \in [r_2]} \gamma_i(x_1, x_{J_{i, 1}}) \gamma^{\prime}_i(x_2, x_{J_{i, 2}})\gamma^{\prime\prime}_{i, 1}(x_{J^{\prime}_{i, 1}}) \cdots \gamma^{\prime\prime}_{i, d^{\prime}_i}(x_{J^{\prime}_{i, d^{\prime}_i}}).\nonumber\end{align} $$

Step 2. Removing most $\gamma $ forms. In this part of the proof, we perform another iterative procedure in which we keep track of a downset of partitions $\mathcal {P}'\subseteq \mathcal {P}$ with the property that $1$ and $2$ are in different sets in every partition and swapping $1$ and $2$ results in a partition still in $\mathcal {P}'$ and we find another multilinear form $\alpha " \colon G^k \to \mathbb {F}_2$ such that $\operatorname {prank}(\alpha ' + \alpha ") \leq O(\exp ^{(O(1))}(r))$ and we have an equality

(4.8) $$ \begin{align}&\alpha"(x_1, x_2, x_3, \dots, x_k) + \alpha"(x_2, x_1, x_3, \dots, x_k) \\&\quad= \sum_{i \in [s]} \gamma_i(x_1,x_{I_i}) \gamma^{\prime}_i(x_2, x_{I^{\prime}_i}) \delta_{i, 1}(x_{J_{i,1}}) \cdots \delta_{i, d_i}(x_{J_{i,d_i}}),\nonumber\end{align} $$

where $s \leq O(\exp ^{(O(1))}(r))$ and each partition $(\{1\} \cup I_i, \{2\} \cup I^{\prime }_i, J_{i,1}, \dots , J_{i, d_i})$ belongs to $\mathcal {P}'$ . (We misuse the notation as the multilinear forms $\gamma _i, \gamma ^{\prime }_i$ are not necessarily identical to those in the assumed decomposition (4.7) and are being modified in each step of the procedure.) Initially, we put $\alpha " = \alpha '$ . We now describe a step in the procedure. Let $\Big \{\{1\} \cup I,\{2\} \cup I', J_1, \dots , J_d\Big \}$ be a maximal partition in $\mathcal {P}'$ , where the priority is given to partitions with $I \cup I' \not =\emptyset $ . Let $\mathcal {P}"$ be the downset of partitions obtained by removing $\Big \{\{1\} \cup I,\{2\} \cup I', J_1, \dots , J_d\Big \}$ and $\Big \{\{1\} \cup I',\{2\} \cup I, J_1, \dots , J_d\Big \}$ from $\mathcal {P}'$ . Assume first that $I \cup I' \not =\emptyset $ .

Before proceeding, we need to regularize the forms appearing in the expression above. We consider the following $d+2$ lists of forms:

• • $\gamma _i(u, x_I)$ for all $i \in [s]$ such that $I_i = I$ and $\gamma ^{\prime }_i(u, x_I)$ for all $i \in [s]$ such that $I^{\prime }_i = I$ .

• • $\gamma _i(u, x_I)$ for all $i \in [s]$ such that $I_i = I'$ and $\gamma ^{\prime }_i(u, x_I)$ for all $i \in [s]$ such that $I^{\prime }_i = I'$ .

• • For each $j \in [d]$ , make the list of all forms that depend on the variables $x_{J_j}$ .

Let $C, D \geq 2$ be constants to be chosen later. Apply the $m = 0$ case of Lemma 4.1 (see Remark 4.2) to each of these lists. We thus obtain further lists of forms:

• • $\phi _i(u, x_I)$ for $i \in [t_1]$ where $t_1 \leq 2s$ ,

• • $\psi _i(u, x_{I'})$ for $i \in [t_2]$ where $t_2 \leq 2s$ , and

• • for each $j \in [d]$ , a list $\mu _{j,i}(x_{J_j})$ with $i \in [q_j]$ for some $q_j \leq s$ ,

and a quantity $R \leq C^{D^{O_k(s)}}$ with properties:

• • each multilinear form in one of the initial $d+2$ lists can be expressed as a sum of a linear combination of forms in the corresponding new list and a multilinear form of partition rank at most R,

• • for each of $d+2$ new lists, the nonzero linear combinations have partition rank at least $(C(R + 2s))^D$ .

Replacing the old forms by the new ones, we have scalars $\lambda _{i_1, i_2, j_1, \dots , j_d}, \lambda ^{\prime }_{i_1, i_2, j_1, \dots , j_d}$ , where $i_1 \in [t_1], i_2 \in [t_2], j_1 \in [q_1], \dots , j_d \in [q_d]$ , and a multilinear form $L(x_{[k]})$ which is a sum of at most $O(s(s + R)^k)$ products whose partitions lie in $\mathcal {P}"$ such that

(4.9) $$ \begin{align}&\alpha"(x_1, x_2, x_3, \dots, x_k) + \alpha"(x_2, x_1, x_3, \dots, x_k)\nonumber\\ &\quad= \sum_{i_1 \in [t_1], i_2 \in [t_2], j_1 \in [q_1], \dots, j_d \in [q_d]} \lambda_{i_1, i_2, j_1, \dots, j_d} \phi_{i_1}(x_1,x_{I}) \psi_{i_2}(x_2, x_{I'}) \mu_{1, j_1}(x_{J_1}) \cdots \mu_{d, j_d}(x_{J_d}) \nonumber\\ &\quad+ \sum_{i_1 \in [t_1], i_2 \in [t_2], j_1 \in [q_1], \dots, j_d \in [q_d]} \lambda^{\prime}_{i_1, i_2, j_1, \dots, j_d} \phi_{i_1}(x_2,x_{I}) \psi_{i_2}(x_1, x_{I'}) \mu_{1, j_1}(x_{J_1}) \cdots \mu_{d, j_d}(x_{J_d}) + L(x_{[k]}).\end{align} $$

Using the fact that

(4.10) $$ \begin{align}\Big(\alpha"(x_1, x_2, x_{[3,k]}) + \alpha"(x_2, x_1, x_{[3,k]})\Big) + \Big(\alpha"(x_1, x_2, x_{[3,k]}) + \alpha"(x_2, x_1, x_{[3,k]})\Big)\circ(1\,\,2) = 0.\end{align} $$

Lemma 3.1 implies $\lambda _{i_1, i_2, j_1, \dots , j_d} = \lambda ^{\prime }_{i_1, i_2, j_1, \dots , j_d}$ , provided C and D are sufficiently large compared only to the constants from Lemma 3.1 and k. Thus, setting

$$\begin{align*}&\alpha"'(x_{[k]}) = \alpha"(x_{[k]}) \\&\quad+ \sum_{i_1 \in [t_1], i_2 \in [t_2], j_1 \in [q_1], \dots, j_d \in [q_d]} \lambda_{i_1, i_2, j_1, \dots, j_d} \phi_{i_1}(x_1,x_{I}) \psi_{i_2}(x_2, x_{I'}) \mu_{1, j_1}(x_{J_1}) \cdots \mu_{d, j_d}(x_{J_d}),\end{align*}$$

it follows that $\operatorname {prank}(\alpha " + \alpha "') \leq O(\exp ^{(O(1))}(r))$ and $\alpha "' + \alpha "' \circ (1\,\,2)$ satisfies (4.8) for the downset of partitions $\mathcal {P}"$ and $s \leq O(\exp ^{(O(1))}(r))$ .

Step 3. Removing remaining forms. Let us now treat the case $I = I' = \emptyset $ . The same steps of the argument apply, except that this time the first two of the $d+2$ lists of forms become the same list of linear forms $\gamma _i(u)$ and $\gamma ^{\prime }_i(u)$ for $i \in [s]$ and we may omit the list of regularized forms $\psi _i$ . This time, we have that

$$ \begin{align*}&\alpha"'(x_1, x_2, x_3, \dots, x_k) + \alpha"'(x_2, x_1, x_3, \dots, x_k)\nonumber\\ &\quad= \sum_{i \in [t_1], j_1 \in [q_1], \dots, j_d \in [q_d]} \lambda_{i_1, i_2, j_1, \dots, j_d} \phi_{i_1}(x_1) \phi_{i_2}(x_2) \mu_{1, j_1}(x_{J_1}) \cdots \mu_{d, j_d}(x_{J_d}) + L(x_{[k]}).\end{align*} $$

Let us define the subspace $U = \{u \in G \colon (\forall i \in [t_1]) \phi _i(u) = 0\}$ , which has a codimension at most $O(\exp ^{(O(1))}(r))$ in G. The expression above implies that $\alpha "'$ is symmetric in the first two variables on the subspace U. Let $\pi \colon G \to U$ be an arbitrary linear projection onto U. Set $\tilde {\alpha }(x_1, \dots , x_k) = \alpha "'(\pi (x_1), \dots , \pi (x_k))$ . Then the form $\tilde {\alpha }$ is symmetric in the first two variables, and by Lemma 2.5, we have $\operatorname {prank}(\alpha "' + \tilde {\alpha }) \leq k \operatorname {codim}_G U$ , implying $\operatorname {prank}(\alpha + \tilde {\alpha }) \leq O(\exp ^{(O(1))}(r))$ , as desired.

Proof Simple algebraic manipulation relying on the symmetry of $\alpha $ gives

$$ \begin{align*}\phi + \phi \circ (1\,\,3) + \phi\circ(1\,\,4) =\ & \alpha + \alpha \circ (3\,\,4) + \alpha \circ (1\,\,3) + \alpha \circ (3\,\,4) \circ (1\,\,3) \\ &\hspace{4cm} + \alpha\circ(1\,\,4) + \alpha \circ (3\,\,4)\circ(1\,\,4) \\ =\ &\alpha + \alpha \circ (3\,\,4) + \alpha + \alpha \circ (1\,\,4) + \alpha\circ(1\,\,4) + \alpha \circ (3\,\,4)\\ =\ & 0.\end{align*} $$

If $\phi $ is symmetric in variables $x_1$ and $x_{4}$ , then we get

$$ \begin{align*}0 = \phi + \phi \circ (1\,\,3) + \phi\circ(1\,\,4) = \phi \circ (1\,\,3) =\ &\alpha \circ (1\,\,3) + \alpha \circ (3\,\,4) \circ (1\,\,3)\\ =\ &\alpha + \alpha \circ (1\,\,4).\end{align*} $$

As $\alpha $ is already symmetric in variables $x_{[3]}$ , it follows that it is symmetric in $x_{[4]}$ , as required.

Proof of Theorem 1.7

By assumptions, we have that

(4.11) $$ \begin{align}\alpha(x_{[4]})+ \alpha \circ (3\,\,4)(x_{[4]}) = \sum_{i \in [r]} \beta_{i, 1}(x_{I_{i,1}}) \beta_{i, 2}(x_{I_{i,2}}) \dots \beta_{i, d_i}(x_{I_{i,d_i}})\end{align} $$

for some multilinear forms $\beta _{i, j}$ such that, for each $i \in [r]$ , the sets $I_{i,1}, \dots , I_{i,d_i}$ form a partition of $[4]$ , and $d_i \geq 2$ . Notice that, for a given $i \in [r]$ , either we have that some $I_{i,j}$ is a singleton, or we have $d_i = 2$ and both sets $I_{i,1}$ and $I_{i,2}$ have size 2. Let $\mathcal {I}$ be the set of indices $i \in [r]$ where we get a singleton set $I_{i,j(i)}$ for some $j(i) \leq d_i$ . If we set $U = \{u \in G \colon \beta _{i, j(i)}(u) = 0\}$ , then we have for all $x_1, x_2, x_3, x_4 \in U$ that

(4.12) $$ \begin{align}\alpha(x_{[4]})+ \alpha \circ (3\,\,4)(x_{[4]}) = \sum_{i \in [r] \setminus \mathcal{I}} \beta_{i, 1}(x_{I_{i,1}}) \beta_{i, 2}(x_{I_{i,2}})\end{align} $$

holds and $|I_{i,1}| = |I_{i,2}| = 2$ for all indices i appearing in the displayed equality. Take the list of all bilinear forms appearing on the right-hand side of (4.12) and apply Lemma 4.1 to it, with parameters $C, D$ to be chosen later (which will depend only on constants in Lemma 3.1) and $m = 1$ . We get a positive integer $R \leq C^{D^{O(r)}}$ , bilinear forms $\sigma _1, \dots , \sigma _s, \rho _1, \dots , \rho _t$ , where $s + t \leq r$ , such that:

1. (i) we have $\operatorname {rank}\Big ( \sigma _i + \sigma _i \circ (1\,\,2)\Big ) \leq R$ for all $i \in [s]$ ,

2. (ii) all nonzero linear combinations of forms $\sigma _1, \dots , \sigma _s, \rho _1, \dots , \rho _t$ and $ \rho _1 \circ (1\,\,2), \dots , \rho _t\circ (1\,\,2)$ have rank at least $(C(R + 2r))^D$ , and

3. (iii) every bilinear form in the list differs from a linear combination of forms $\sigma _{[s]}, \rho _{[t]}$ and $ \rho _{[t]} \circ (1\,\,2)$ by a bilinear form of rank at most R.

Passing to a further subspace $U' \leq U$ of codimension at most $2rR$ in U, we may assume that properties (i) and (iii) are exact rather than approximate, i.e., $\sigma _i = \sigma _i \circ (1\,\,2)$ and every bilinear form in the list equals a linear combination of the newly found forms. We may replace the bilinear forms in (4.12) with the newly found forms. This leads to equality

(4.13) $$ \begin{align}&\alpha(x_{[4]})+ \alpha \circ (3\,\,4)(x_{[4]}) \nonumber\\ &\hspace{1cm}= \sum_{\substack{i \in [s]\\(a,b,c,d) \in \mathcal{V}_1}} \lambda_{\substack{i\\a,b,c,d}} \sigma_i(x_a, x_b) \sigma_i(x_c, x_d) + \sum_{\substack{1\leq i < j \leq s\\(a,b,c,d) \in \mathcal{V}_2}} \lambda_{\substack{i,j\\a,b,c,d}} \sigma_i(x_a, x_b) \sigma_j(x_c, x_d)\nonumber\\ &\hspace{2cm} + \sum_{\substack{i \in [t]\\(a,b,c,d) \in \mathcal{V}_3}} \!\mu_{\substack{i\\a,b,c,d}} \rho_i(x_a, x_b) \rho_i(x_c, x_d) + \sum_{\substack{1\leq i < j \leq t\\(a,b,c,d) \in \mathcal{V}_4}} \!\mu_{\substack{i,j\\a,b,c,d}} \rho_i(x_a, x_b) \rho_j(x_c, x_d)\nonumber\\ &\hspace{2cm} + \sum_{\substack{i \in [s],j \in [t]\\(a,b,c,d) \in \mathcal{V}_5}} \nu_{\substack{i,j\\a,b,c,d}} \sigma_i(x_a, x_b) \rho_j(x_c, x_d),\end{align} $$

for all $x_1, x_2, x_3, x_4 \in U'$ , where:

• • $\mathcal {V}_1 = \{(1,2,3,4), (1,3,2,4), (1,4,2,3)\}$ ,

• • $\mathcal {V}_2 = \{(1,2,3,4), (1,3,2,4), (1,4,2,3), (2,3,1,4), (2,4,1,3), (3,4,1,2)\}$ ,

• • $\mathcal {V}_3 = \{(1,2,3,4), (2,1,3,4), (1,2,4,3), (2,1,4,3), (1,3,2,4), (3,1,2,4), (1,3,4,2), (3,1,4,2), (1,4,2,3), (4,1,2,3), (1,4,3,2), (4,1,3,2)\}$ ,

• • $\mathcal {V}_4 = \operatorname {Sym}_{[4]}$ , and

• • $\mathcal {V}_5 = \{(1,2,3,4),(1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2), (2,3,1,4), (2,3,4,1), (2,4,1,3), (2,4,3,1), (3,4,1,2), (3,4,2,1)\}$ .

Write $\phi (x_{[4]}) = \alpha (x_{[4]})+ \alpha \circ (3\,\,4)(x_{[4]})$ . This form has the following properties:

1. (i) $\phi = \phi \circ (1\,\,2)$ .

2. (ii) $\phi = \phi \circ (3\,\,4)$ .

3. (iii) $\phi + \phi \circ (1\,\,3) + \phi \circ (1\,\,4) = 0$ .

4. (iv) $\phi (u,v,u,v) = 0$ for all $u,v \in U'$ .

Let us briefly explain why $\phi $ has the claimed properties. Property (i) follows from the fact that $\alpha $ is symmetric in the first two variables. Property (ii) follows directly from the definition of $\phi $ . Property (iii) follows from Lemma 4.4. Finally, property (iv) follows from assumptions (i) and (ii) in the statement of Theorem 1.7, namely we have

$$ \begin{align*}\phi(u,v,u,v) =\ &\alpha(u,v,u,v) + \alpha(u,v,v,u)\hspace{2cm}(\text{definition of }\phi)\\ =\ &\alpha(u,u,v,v) + \alpha(v,v,u,u)\hspace{2cm}(\text{symmetry of }\alpha)\\ =\ &0\hspace{2cm}(\text{Theorem~1.7, assumption {(ii)}}).\end{align*} $$

We use these properties of $\phi $ to make $\alpha $ symmetric in all variables. Note that applying Lemma 3.1 to the sum $\phi + \phi \circ (1\,\,2)$ , which vanishes by symmetry property (i), implies that we have equality $\lambda _{\substack {i\\1,3,2,4}} = \lambda _{\substack {i\\1,4,2,3}}$ . Using Lemma 3.1 similarly for either of the symmetry properties (i) and (ii) and for other coefficients, we obtain additional equalities, such as $\mu _{\substack {i\\1,2,3,4}} = \mu _{\substack {i\\2,1,3,4}}$ . In the rest of the proof, we obtain further identities between coefficients, which are not direct consequences of symmetry properties (i) and (ii). We treat each of the five sums in (4.13) separately. Once we get sufficient information on relationships between different coefficients we shall be able to modify $\alpha $ by explicit multilinear forms of small bounded partition rank and end up with an exactly symmetric form.

Let us also stress that the additional assumption (ii) in Theorem 1.7 that makes it possible to pass from approximately symmetric form to an exactly symmetric one is used in Step 1 and Step 2 of the proof below. More precisely, in the proof itself, we use the property (iv) of the multilinear form $\phi $ , which uses the mentioned additional property of $\alpha $ .

Step 1. Coefficients $\lambda _{\substack {i\\a,b,c,d}}$ . Fix any $i \in [s]$ . To see that $\lambda _{\substack {i\\1,2,3,4}} = 0$ , observethat the coefficient of $\sigma _i(x_1, x_2)\sigma _i(x_3, x_4)$ in the expression $\phi + \phi \circ (1\,\,3) + \phi \circ (1\,\,4)$ equals

$$\begin{align*}\lambda_{\substack{i\\1,2,3,4}} + \lambda_{\substack{i\\1,4,2,3}} + \lambda_{\substack{i\\1,3,2,4}}.\end{align*}$$

By property (iii) of $\phi $ and Lemma 3.1, we conclude that

$$\begin{align*}\lambda_{\substack{i\\1,2,3,4}} + \lambda_{\substack{i\\1,4,2,3}} + \lambda_{\substack{i\\1,3,2,4}} = 0.\end{align*}$$

Since $\lambda _{\substack {i\\1,4,2,3}} = \lambda _{\substack {i\\1,3,2,4}}$ from discussion before this step, we finally get $\lambda _{\substack {i\\1,2,3,4}} = 0$ .

Next, we show that $\lambda _{\substack {i\\1,4,2,3}} = \lambda _{\substack {i\\1,3,2,4}} = 0$ . To that end, let $X = \{u \in U' \colon (\forall i \in [s]) \sigma _i(u,u) = 0, (\forall i \in [t]) \rho _i(u,u) = 0\}$ . By Lemma 2.2, this set has size $|X| \geq 2^{-s-t-1}|U'|$ . Since all nonzero linear combinations of forms $\sigma _{[s]}$ , $\rho _{[t]}$ and $\rho _{[t]}\circ (1\,\,2)$ have rank at least $(C(R + 2r))^D - 2rR$ on $U'$ , we may find $u,v \in X$ such that $\sigma _i(u,v) = 1$ , $\sigma _j(u,v) = 0$ for all $j \not =i$ and $\rho _j(u,v) = \rho _j(v,u) = 0$ for all $j \in [t]$ . Putting $x_1 = x_3 = u, x_2 = x_4 = v$ in (4.13), we deduce that

$$\begin{align*}0 = \phi(u,v,u,v) = \lambda_{\substack{i\\1,3,2,4}} \sigma_i(u,u)\sigma_i(v,v) + \lambda_{\substack{i\\1,4,2,3}} \sigma_i(u,v)\sigma_i(u,v) = \lambda_{\substack{i\\1,4,2,3}},\end{align*}$$

as required, where we used property (iv) of $\phi $ .

Step 2. Coefficients $\mu _{\substack {i\\a,b,c,d}}$ . As remarked before the first step, from symmetry properties (i) and (ii) of $\phi $ and Lemma 3.1, we have scalars $\overline {\mu }_i, \overline {\mu }^{\prime }_i, \overline {\mu }^{\prime \prime }_i, \overline {\mu }^{\prime \prime \prime }_i$ such that

$$ \begin{align*}\overline{\mu}_i &= \mu_{\substack{i\\1,2,3,4}} = \mu_{\substack{i\\2,1,3,4}} = \mu_{\substack{i\\1,2,4,3}} = \mu_{\substack{i\\2,1,4,3}},\\ \overline{\mu}^{\prime}_i &= \mu_{\substack{i\\3,1,2,4}} = \mu_{\substack{i\\1,4,3,2}} = \mu_{\substack{i\\4,1,2,3}} = \mu_{\substack{i\\1,3,4,2}},\\ \overline{\mu}^{\prime\prime}_i &= \mu_{\substack{i\\1,3,2,4}} = \mu_{\substack{i\\1,4,2,3}}\,\,\text{and}\,\,\,\, \overline{\mu}^{\prime\prime\prime}_i= \mu_{\substack{i\\3,1,4,2}} = \mu_{\substack{i\\4,1, 3,2}}.\end{align*} $$

Fix any $i \in [t]$ . We now show that $\overline {\mu }_i + \overline {\mu }^{\prime }_i + \overline {\mu }^{\prime \prime }_i = 0$ and $\overline {\mu }_i + \overline {\mu }^{\prime }_i + \overline {\mu }^{\prime \prime \prime }_i = 0$ . Observe that the coefficient of $\rho _i(x_3, x_1)\rho _i(x_4, x_2)$ in the expression $\phi + \phi \circ (1\,\,3) + \phi \circ (1\,\,4)$ equals

$$\begin{align*}\mu_{\substack{i\\3,1,4,2}} + \mu_{\substack{i\\1,3,4,2}} + \mu_{\substack{i\\1,2,3,4}} = \overline{\mu}^{\prime\prime\prime}_i + \overline{\mu}^{\prime}_i + \overline{\mu}_i.\end{align*}$$

By Lemma 3.1, we see that $\overline {\mu }_i + \overline {\mu }^{\prime }_i + \overline {\mu }^{\prime \prime \prime }_i = 0$ . Similarly, we consider the coefficient of $\rho _i(x_1, x_3)\rho _i(x_2, x_4)$ in the expression $\phi + \phi \circ (1\,\,3) + \phi \circ (1\,\,4)$ , which equals

$$\begin{align*}\mu_{\substack{i\\1,3,2,4}} + \mu_{\substack{i\\3,1,2,4}} + \mu_{\substack{i\\2,1,4,3}} = \overline{\mu}^{\prime\prime}_i + \overline{\mu}^{\prime}_i + \overline{\mu}_i,\end{align*}$$

which vanishes by Lemma 3.1. As the last equality between coefficients in this step, we claim that $\overline {\mu }_i = \overline {\mu }^{\prime }_i$ . Let $X = \{u \in U' \colon (\forall i \in [s]) \sigma _i(u,u) = 0, (\forall i \in [t]) \rho _i(u,u) = 0\}$ . By Lemma 2.2, this set has size $|X| \geq 2^{-s-t-1}|U'|$ . We thus have a choice of $u,v \in X$ such that $\rho _i(u,v) = 1$ , $\rho _i(v,u) = 0$ , $\rho _j(u,v) = \rho _j(u,v) = 0$ for $j \not = i$ and $\sigma _j(u,v) = \sigma _j(v,u) = 0$ for $j \in [t]$ . Then, setting $x_1 = x_3 = u, x_2 = x_4 = v$ , we get

$$\begin{align*}0 = \phi(u,v,u,v) &= \sum_{(a,b,c,d) \in \mathcal{V}_3} \mu_{\substack{i\\a,b,c,d}} \rho_i(x_a, x_b) \rho_i(x_c, x_d) \\&= \overline{\mu}_i \rho_i(u,v)\rho_i(u,v) + \overline{\mu}^{\prime}_i \rho_i(u,v)\rho_i(u,v) = \overline{\mu}_i + \overline{\mu}^{\prime}_i ,\end{align*}$$

where we used property (iv) of $\phi $ . Thus, we have $\overline {\mu }^{\prime }_i = \overline {\mu }_i$ and $\overline {\mu }^{\prime \prime }_i = \overline {\mu }^{\prime \prime \prime }_i = 0$ .

Define a multilinear form $\alpha ^{(1)} \colon G^4 \to \mathbb {F}_2$ by

$$\begin{align*}\alpha^{(1)}(x_1, x_2, x_3, x_4) = \alpha(x_1, x_2, x_3, x_4) + \sum_{i \in [t]} \overline{\mu}_i \Big(\sum_{\pi \in \operatorname{Sym}_{[3]}}\rho_i(x_{\pi_1}, x_{\pi_2})\rho_i(x_{\pi_3}, x_4)\Big).\end{align*}$$

By our work in the first two steps, we conclude that

(4.14) $$ \begin{align}\alpha^{(1)}(x_{[4]})+ \alpha^{(1)} \circ (3\,\,4)(x_{[4]}) = &\sum_{\substack{1\leq i < j \leq s\\(a,b,c,d) \in \mathcal{V}_2}} \lambda_{\substack{i,j\\a,b,c,d}} \sigma_i(x_a, x_b) \sigma_j(x_c, x_d) \\&+ \sum_{\substack{1\leq i < j \leq t\\(a,b,c,d) \in \mathcal{V}_4}} \mu_{\substack{i,j\\a,b,c,d}} \rho_i(x_a, x_b) \rho_j(x_c, x_d)\nonumber\\ & + \sum_{\substack{i \in [s],j \in [t]\\(a,b,c,d) \in \mathcal{V}_5}} \nu_{\substack{i,j\\a,b,c,d}} \sigma_i(x_a, x_b) \rho_j(x_c, x_d)\nonumber\end{align} $$

holds for all $x_1, x_2, x_3, x_4 \in U'$ . Let us set $\phi ^{(1)} = \alpha ^{(1)} + \alpha ^{(1)}\circ (3\,\,4)$ . Since $\sum _{\pi \in \operatorname {Sym}_{[3]}}\rho _i(x_{\pi _1}, x_{\pi _2})\rho _i(x_{\pi _3}, x_4)$ is symmetric in variables $x_{[3]}$ , the form $\phi ^{(1)}$ still satisfies symmetry conditions (i)–(iii) which we used for $\phi $ .

Step 3. Coefficients $\lambda _{\substack {i,j\\a,b,c,d}}$ . As discussed prior to the first step, by symmetry properties (i) and (ii) of $\phi $ and Lemma 3.1, we know that the coefficients $\lambda _{\substack {i,j\\1,3,2,4}}, \lambda _{\substack {i,j\\1,4,2,3}}, \lambda _{\substack {i,j\\2,3,1,4}}$ , and $\lambda _{\substack {i,j\\2,4,1,3}}$ are equal. Write $\overline {\lambda }_{i,j}$ for this value. We show that the remaining coefficients in this step $\lambda _{\substack {i,j\\1,2,3,4}}$ and $\lambda _{\substack {i,j\\3,4,1,2}}$ both vanish.

To see this, consider the coefficients of $\sigma _i(x_1, x_2)\sigma _j(x_3, x_4)$ and $\sigma _i(x_3, x_4)\sigma _j(x_1, x_2)$ in $\phi ^{(1)} + \phi ^{(1)} \circ (1\,\,3) + \phi ^{(1)}\circ (1\,\,4)$ . These are, respectively,

$$\begin{align*}\lambda_{\substack{i,j\\1,2,3,4}} + \lambda_{\substack{i,j\\2,3,1,4}} + \lambda_{\substack{i,j\\2,4,1,3}} = \lambda_{\substack{i,j\\1,2,3,4}}\end{align*}$$

and

$$\begin{align*}\lambda_{\substack{i,j\\3,4,1,2}} + \lambda_{\substack{i,j\\1,4,2,3}} + \lambda_{\substack{i,j\\1,3,2,4}} = \lambda_{\substack{i,j\\3,4,1,2}}.\end{align*}$$

Lemma 3.1 shows that both expressions vanish, showing that $\lambda _{\substack {i,j\\1,2,3,4}} = \lambda _{\substack {i,j\\3,4,1,2}} = 0$ . Define $\alpha ^{(2)} \colon G^4 \to \mathbb {F}_2$ by

$$ \begin{align*}&\alpha^{(2)}(x_1, x_2, x_3, x_4) = \alpha^{(1)}(x_1, x_2, x_3, x_4) \\ &\quad+ \sum_{\substack{i,j \in [s]\\i \not=j}} \overline{\lambda}_{i,j} \Big(\sigma_i(x_1, x_2)\sigma_j(x_3, x_4) + \sigma_i(x_2, x_3)\sigma_j(x_1, x_4)+ \sigma_i(x_3, x_1)\sigma_j(x_2, x_4)\Big).\end{align*} $$

Note that the additional term in the equality above is symmetric in $x_{[3]}$ on the subspace $U'$ and so is $\alpha ^{(2)}(x_{[4]})$ . Defining $\phi ^{(2)}(x_{[4]}) = \alpha ^{(2)}(x_{[4]})+ \alpha ^{(2)} \circ (3\,\,4)(x_{[4]})$ , we observe that $\phi ^{(2)}|_{U' \times \cdots \times U'}$ satisfies symmetry conditions (i)–(iii) which we used for $\phi $ . It follows from the definition of $\alpha ^{(2)}$ that

(4.15) $$ \begin{align}\alpha^{(2)}(x_{[4]})+ \alpha^{(2)} \circ (3\,\,4)(x_{[4]}) = &\sum_{\substack{1\leq i < j \leq t\\(a,b,c,d) \in \mathcal{V}_4}} \mu_{\substack{i,j\\a,b,c,d}} \rho_i(x_a, x_b) \rho_j(x_c, x_d)\nonumber\\ &\hspace{2cm}\!\!\!\!\!\!\!\!\!\! + \sum_{\substack{i \in [s],j \in [t]\\(a,b,c,d) \in \mathcal{V}_5}} \nu_{\substack{i,j\\a,b,c,d}} \sigma_i(x_a, x_b) \rho_j(x_c, x_d)\end{align} $$

holds for all $x_1, x_2, x_3, x_4 \in U'$ .

In order to simplify the notation in the remaining two steps, we introduce informal objects called places. Each symmetric form $\sigma _i$ has one place, and each asymmetric form $\rho _i$ has two places corresponding to the first variable and the second variable. Notice that the symmetry properties (i) and (ii) of $\phi $ and Lemma 3.1 imply that the coefficients of products $\rho _i(x_a, x_b) \rho _j(x_c, x_d)$ and $\sigma _i(x_a, x_b) \rho _j(x_c, x_d)$ depend only on the choice of places for variables $x_3$ and $x_4$ . For example, if we chose the place in $\sigma _i$ and a place corresponding to the second variable in $\rho _j$ , then we have that products

$$\begin{align*}\sigma_i(x_1, x_3) \rho_j(x_2, x_4), \sigma_i(x_2, x_3) \rho_j(x_1, x_4), \sigma_i(x_1, x_4) \rho_j(x_2, x_3), \sigma_i(x_2, x_4) \rho_j(x_1, x_3)\end{align*}$$

receive the same coefficient. Furthermore, our work in Step 3 could be rephrased as saying that the coefficients of products where we use two copies of the same place vanish.

Using the places terminology, we write $\lambda _{A\,\,B}$ for the coefficient received by products where $x_3$ and $x_4$ are put at places A and B. In the rest of the proof, we show that

(4.16) $$ \begin{align}\lambda_{A\,\,B} + \lambda_{A\,\,C} + \lambda_{B\,\,C} = 0\end{align} $$

for all places $A, B, C$ .

Step 4. Coefficients $\mu _{\substack {i,j\\a,b,c,d}}$ . Note that, in this case, the places $A, B$ , and C in (4.16) may assume to be distinct and furthermore that A and B belong to the form $\rho _i$ and C is a place in $\rho _j$ (we do not assume that $i < j$ , but only that $i \not =j$ ). Suppose that C is the place corresponding to the first variable in $\rho _j$ , and the other case will follow analogously. Consider the coefficient of the product $\rho _i(x_3, x_4)\rho _j(x_1, x_2)$ in $\phi ^{(2)} + \phi ^{(2)} \circ (1\,\,3) + \phi ^{(2)}\circ (1\,\,4) = 0$ . This coefficient equals

$$\begin{align*}\lambda_{A\,\,B} + \lambda_{B\,\,C} + \lambda_{A\,\,C}\end{align*}$$

and vanishes, proving (4.16).

For each $i,j \in [t]$ with $i < j$ let $\overline {\mu }_{i,j, A} = \lambda _{A\,\,D}$ , $\overline {\mu }_{i,j, B} = \lambda _{B\,\,D}$ and $\overline {\mu }_{i,j, C} = \lambda _{C\,\,D}$ , where $A, B, C$ , and D are, respectively, the places corresponding to the first variable in $\rho _i$ , the second variable in $\rho _i$ , the first variable in $\rho _j$ , and the second variable in $\rho _j$ . We may now define $\alpha ^{(3)} \colon G^4 \to \mathbb {F}_2$ by

(4.17) $$ \begin{align}\alpha^{(3)}(x_1, x_2, x_3, x_4) =\alpha^{(2)}(x_1, x_2, x_3, x_4) + &\sum_{1 \leq i < j \leq t} \overline{\mu}_{i,j, A}\Big(\sum_{\pi \in \operatorname{Sym}_{[3]}} \rho_i(x_4, x_{\pi(1)}) \rho_j(x_{\pi(2)}, x_{\pi(3)})\Big)\nonumber\\ & + \sum_{1 \leq i < j \leq t} \overline{\mu}_{i,j, B}\Big(\sum_{\pi \in \operatorname{Sym}_{[3]}} \rho_i(x_{\pi(1)}, x_4) \rho_j(x_{\pi(2)}, x_{\pi(3)})\Big)\nonumber\\ & + \sum_{1 \leq i < j \leq t} \overline{\mu}_{i,j, C}\Big(\kern-1pt\sum_{\pi \in \operatorname{Sym}_{[3]}} \rho_i(x_{\pi(1)}, x_{\pi(2)}) \rho_j(x_4, x_{\pi(3)})\kern-1pt\Big).\end{align} $$

It follows that $\alpha ^{(3)}|_{U'\times \cdots \times U'}$ is still symmetric in the first three variables and the form $\phi ^{(3)}(x_{[4]}) = \alpha ^{(3)}(x_{[4]})+ \alpha ^{(3)} \circ (3\,\,4)(x_{[4]})$ satisfies symmetry conditions (i)–(iii) which we used for $\phi $ on the subspace $U'$ . We claim that $\phi ^{(3)}$ has simpler structure than $\phi ^{(2)}$ .

Claim 4.5 The equality

(4.18) $$ \begin{align}\alpha^{(3)}(x_{[4]})+ \alpha^{(3)} \circ (3\,\,4)(x_{[4]}) = \sum_{\substack{i \in [s],j \in [t]\\(a,b,c,d) \in \mathcal{V}_5}} \nu_{\substack{i,j\\a,b,c,d}} \sigma_i(x_a, x_b) \rho_j(x_c, x_d)\end{align} $$

holds for all $x_1, x_2, x_3, x_4 \in U'$ .

Proof of Claim 4.5

Fix $1\leq i < j \leq t$ . Let $A, B, C$ , and D are, respectively, the places corresponding to the first variable in $\rho _i$ , the second variable in $\rho _i$ , the first variable in $\rho _j$ , and the second variable in $\rho _j$ . Let $(a,b,c,d) \in \mathcal {V}_4$ . Recall that the coefficient of $\rho _i(x_a, x_b)\rho _j(x_c, x_d)$ in (4.15) is $\mu _{\substack {i,j\\a,b,c,d}}$ . Our goal is to show that this coefficient equals the one in the form

$$\begin{align*}\Delta(x_{[4]}) = \Big(\alpha^{(3)}(x_{[4]}) + \alpha^{(2)}(x_{[4]})\Big) + \Big(\alpha^{(3)} \circ (3\,\,4)(x_{[4]}) + \alpha^{(2)} \circ (3\,\,4)(x_{[4]})\Big)\end{align*}$$

arising from definition (4.17). We look at different cases on places for $x_3$ and $x_4$ .

Case 1: $x_3,x_4$ are at places $A, B$ . We have $\{a,b\} = \{3,4\}$ and $\{c,d\} = \{1,2\}$ . The coefficient of $\rho _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ equals $\overline {\mu }_{i,j, A} + \overline {\mu }_{i,j, B} = \lambda _{A\,\,D} + \lambda _{B\,\,D} = \lambda _{A\,\,B} = \mu _{\substack {i,j\\a,b,c,d}}$ , where we used (4.16) for places $A, B$ , and D in the second equality.

Case 2: $x_3,x_4$ are at places $A, C$ . We have $\{a,c\} = \{3,4\}$ and $\{b,d\} = \{1,2\}$ . The coefficient of $\rho _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ equals $\overline {\mu }_{i,j, A} + \overline {\mu }_{i,j, C} = \lambda _{A\,\,D} + \lambda _{C\,\,D} = \lambda _{A\,\,C} = \mu _{\substack {i,j\\a,b,c,d}}$ , where we used (4.16) for places $A, C$ , and D in the second equality.

Case 3: $x_3,x_4$ are at places $A, D$ . We have $\{a,d\} = \{3,4\}$ and $\{b,c\} = \{1,2\}$ . The coefficient of $\rho _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ equals $\overline {\mu }_{i,j, A} = \lambda _{A\,\,D} = \mu _{\substack {i,j\\a,b,c,d}}$ .

Case 4: $x_3,x_4$ are at places $B, C$ . We have $\{b,c\} = \{3,4\}$ and $\{a,d\} = \{1,2\}$ . The coefficient of $\rho _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ equals $\overline {\mu }_{i,j, B} + \overline {\mu }_{i,j, C} = \lambda _{B\,\,D} + \lambda _{C\,\,D} = \lambda _{B\,\,C} = \mu _{\substack {i,j\\a,b,c,d}}$ , where we used (4.16) for places $B, C$ , and D in the second equality.

Case 5: $x_3,x_4$ are at places $B, D$ . We have $\{b,d\} = \{3,4\}$ and $\{a,c\} = \{1,2\}$ . The coefficient of $\rho _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ equals $\overline {\mu }_{i,j, B} = \lambda _{B\,\,D} = \mu _{\substack {i,j\\a,b,c,d}}$ .

Case 6: $x_3,x_4$ are at places $C, D$ . We have $\{c,d\} = \{3,4\}$ and $\{a,b\} = \{1,2\}$ . The coefficient of $\rho _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ equals $\overline {\mu }_{i,j, C} = \lambda _{C\,\,D} = \mu _{\substack {i,j\\a,b,c,d}}$ .

Step 5. Coefficients $\nu _{\substack {i,j\\a,b,c,d}}$ . If two of the places $A, B$ , and C in (4.16) are the same, then without loss of the generality $A= B$ is the place in a symmetric form $\sigma _i$ and C is one of the places in an asymmetric form $\rho _j$ . Assume that C corresponds to the first variable in $\rho _j$ , and the other case is similar. Equality (4.16) reduces to showing that $\lambda _{A\,\,A} = 0$ in this case. To see this, consider the coefficient of the product $\sigma _i(x_3, x_4)\rho _j(x_1, x_2)$ in $\phi ^{(3)} + \phi ^{(3)} \circ (1\,\,3) + \phi ^{(3)}\circ (1\,\,4) = 0$ . This coefficient equals

$$\begin{align*}\lambda_{A\,\,A} + \lambda_{A\,\,C} + \lambda_{A\,\,C}\end{align*}$$

and vanishes, as desired.

On the other hand, if $A, B$ , and C are different, then we may assume that A is the place in $\sigma _i$ and B and C are, respectively, the first variable and the second variable in $\rho _j$ . This time consider the coefficient of the product $\sigma _i(x_3, x_2)\rho _j(x_4, x_1)$ in $\phi ^{(3)} + \phi ^{(3)} \circ (1\,\,3) + \phi ^{(3)}\circ (1\,\,4) = 0$ . This coefficient equals

$$\begin{align*}\lambda_{A\,\,B} + \lambda_{B\,\,C} + \lambda_{A\,\,C}\end{align*}$$

and vanishes, as desired.

For each $i \in [s],j \in [t]$ , let $\overline {\mu }_{i,j, B} = \lambda _{A\,\,B}$ and $\overline {\mu }_{i,j, C} = \lambda _{A\,\,C}$ , where $A, B$ , and C are, respectively, the places corresponding to $\sigma _i$ , the first variable in $\rho _j$ , and the second variable in $\rho _j$ . We may now define $\alpha ^{(4)} \colon G^4 \to \mathbb {F}_2$ by

$$ \begin{align*}&\alpha^{(4)}(x_1, x_2, x_3, x_4) =\alpha^{(3)}(x_1, x_2, x_3, x_4) \\ &\quad+ \sum_{i \in [s], j \in [t]} \overline{\mu}_{i,j, B}\Big(\sigma_i(x_1, x_2)\rho_j(x_4, x_3) + \sigma_i(x_2, x_3)\rho_j(x_4, x_1) + \sigma_i(x_3, x_1)\rho_j(x_4, x_2)\Big)\\ &\quad+ \sum_{i \in [s], j \in [t]} \overline{\mu}_{i,j, C}\Big(\sigma_i(x_1, x_2)\rho_j(x_3, x_4) + \sigma_i(x_2, x_3)\rho_j(x_1, x_4) + \sigma_i(x_3, x_1)\rho_j(x_2, x_4)\Big).\end{align*} $$

We show that $\alpha ^{(4)}|_{U'\times \cdots \times U'}$ is symmetric.

Proof of Claim 4.6

Let us reuse the notation $\Delta $ , this time defining it as

$$\begin{align*}\Delta(x_{[4]}) = \Big(\alpha^{(4)}(x_{[4]}) + \alpha^{(3)}(x_{[4]})\Big) + \Big(\alpha^{(4)} \circ (3\,\,4)(x_{[4]}) + \alpha^{(3)} \circ (3\,\,4)(x_{[4]})\Big).\end{align*}$$

Let $i \in [s], j \in [t]$ , and $(a,b,c,d) \in \mathcal {V}_5$ . We need to show that the coefficient of $ \sigma _i(x_a, x_b) \rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ equals $\nu _{\substack {i,j\\a,b,c,d}} $ . Let $A, B$ , and C be, respectively, the places corresponding to $\sigma _i$ , the first variable in $\rho _j$ , and the second variable in $\rho _j$ . As in the proof of Claim 4.5, we consider the cases for the places of $x_3$ and $x_4$ .

Case 1: $x_3,x_4$ are both at place A. We have $\{a,b\} = \{3,4\}$ and $\{c,d\} = \{1,2\}$ . The coefficient of $\sigma _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ vanishes, and we know that $0 = \lambda _{A\,A} = \nu _{\substack {i,j\\a,b,c,d}}$ , as desired.

Case 2: $x_3,x_4$ are at places $A, B$ . We have $\{b,c\} = \{3,4\}$ and $\{a,d\} = \{1,2\}$ (recalling that $\mathcal {V}_5$ is a proper subset of $\operatorname {Sym}_{[4]}$ ). The coefficient of $\sigma _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ is $\overline {\mu }_{i,j, B} = \lambda _{A\,\,B} = \nu _{\substack {i,j\\a,b,c,d}}$ .

Case 3: $x_3,x_4$ are at places $A, C$ . We have $\{b,d\} = \{3,4\}$ and $\{a,c\} = \{1,2\}$ (again, recalling that $\mathcal {V}_5$ is a proper subset of $\operatorname {Sym}_{[4]}$ ). The coefficient of $\sigma _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ is $\overline {\mu }_{i,j, C} = \lambda _{A\,\,C} = \nu _{\substack {i,j\\a,b,c,d}}$ .

Case 4: $x_3,x_4$ are at places $B, C$ . We have $\{c,d\} = \{3,4\}$ and $\{a,b\} = \{1,2\}$ . The coefficient of $\sigma _i(x_a, x_b)\rho _j(x_c, x_d)$ in $\Delta (x_{[4]})$ is $\overline {\mu }_{i,j, B} + \overline {\mu }_{i,j, C} = \lambda _{A\,\,B} + \lambda _{A\,\,C}= \lambda _{B\,\,C}= \nu _{\substack {i,j\\a,b,c,d}}$ , where we used (4.16) in the second equality.

We have thus obtained a multilinear form $\alpha ^{(4)}|_{U' \times \cdots \times U'}$ which is symmetric in $U' \times \cdots \times U'$ such that $\operatorname {prank}\Big ((\alpha + \alpha ^{(4)})|_{U' \times \cdots \times U'}\Big ) \leq O(r^2)$ . The claimed result follows from Corollary 2.6 (recall that $\operatorname {codim}_G U' \leq O(Rr)$ , which is the main contribution to the final bound). As in the proof of the previous theorem, we choose C and D depending only on the constants from Lemma 3.1 so that the partition rank condition of Lemma 3.1 holds at all times.

Proof of Theorem 1.8

By assumptions, we have that

(4.19) $$ \begin{align}\alpha(x_{[5]})+ \alpha \circ (3\,\,4)(x_{[5]}) = \sum_{i \in [r]} \beta_{i, 1}(x_{I_{i,1}}) \beta_{i, 2}(x_{I_{i,2}}) \dots \beta_{i, d_i}(x_{I_{i,d_i}}),\end{align} $$

where all partitions of the variables appearing above are nontrivial, that is, $d_i \geq 2$ .

Step 1. Obtaining a decomposition with inherited properties. We begin our work by deducing some properties of the low partition rank decomposition (4.19) of $\alpha + \alpha \circ (3\,\,4)$ . Using Proposition 3.3, we may assume that every $\beta _{i, j}$ comes from a slice of $\alpha + \alpha \circ (3\,\,4)$ . This has the cost of replacing r by $O({r}^{O(1)})$ , which we do with a slight misuse of notation. Writing $\beta (x_I)$ for $\beta _{i, j}(x_{I_{i,j}})$ , we know that $\beta (x_I) = \alpha (x_I, y_{[5] \setminus I}) + \alpha \circ (3\,\,4)(x_I, y_{[5] \setminus I})$ for some fixed $y_{[5] \setminus I}$ . In particular, the following holds.

• (S1) When $3, 4 \in I$ , then $\beta (x_I) = \alpha _{y_{[5] \setminus I}}(x_I) + \alpha _{y_{[5] \setminus I}}\circ (3\,\,4)(x_I)$ . Thus, we have the equality $\beta = \tilde {\beta } + \tilde {\beta } \circ (3\,\,4)$ , where $\tilde {\beta }(x_I)$ is a multilinear form which is symmetric in the variables $x_{I \cap [3]}$ . Therefore, replacing $\beta $ by $\tilde {\beta } + \tilde {\beta } \circ (3\,\,4)$ , we may assume that every form $\beta (x_I)$ that has $3, 4 \in I$ is either symmetric in variables $x_{I \cap [3]}$ or symmetric in variables $x_{I \cap (\{1,2,4\})}$ .

• (S2) When $|I \cap \{3, 4\}| = 1$ , say $\{3\} = I \cap \{3, 4\}$ , then we see that $\beta $ is symmetric in variables $x_{I\cap [2]}$ .

• (S3) When $3, 4 \notin I$ , then $\beta $ is symmetric in variables $x_{I \cap [2]}$ .

We may therefore assume that all forms in the products in (4.19) have a symmetry property described in one of (S1)–(S3).

Step 2. Simplifying the partitions. Observe that all nontrivial partitions of the set $[5]$ have either two sets which have sizes 3 and 2, or at least one set is a singleton. We refer to the former partitions as the relevant ones. In particular, each summand in (4.19) either has a relevant partition or it has a factor $\gamma (x_c)$ for some linear form $\gamma $ . By passing to subspace defined by zero sets of such linear forms, we may assume that all partitions are in fact relevant.

Step 3. Regularizing the forms. In this step, we apply our symmetry-respecting regularity lemma (Lemma 4.1).

Our goal is to remove some partitions of variables from expression (4.19). We shall first remove all relevant partitions $I_1 \cup I_2 = [5]$ such that $|I_1| = 2$ and $5 \in I_1$ , and second we remove all relevant partitions $I_1 \cup I_2 = [5]$ such that $|I_1| = 3$ and $5 \in I_1$ . Using slightly more general notation, we explain how to apply Lemma 4.1 in either of the two cases. We say that the variables $x_1, \dots , x_4$ are active and that $x_5$ is the passive variable.

Let $(\ell _1, \ell _2) \in \{(1,3), (2,2)\}$ . The numbers $\ell _1$ and $\ell _2$ stand for the number of active variables in sets $I_1$ and $I_2$ . We aim to remove the set $\mathcal {R}$ of all partitions $(A_1 \cup \{5\}, A_2)$ , where $|A_i| = \ell _i$ . Write $P_1 = \{5\}$ and $P_2 = \emptyset $ , which stand for the indices of passive variables, and make the following list. Take all forms $\beta (x_{A \cup P_i})$ appearing in the decomposition (4.19) such that $A \subseteq [4]$ of size $|A| = \ell _i$ . By the symmetry properties S1–S3, we see that each of the chosen forms $\beta $ is equal to:

• • $\gamma (x_S, x_{P_i})$ where $\gamma $ is symmetric in $x_S$ , $|S| = \ell _i$ , with $S \subseteq [2]$ , or

• • $\gamma (x_S, x_c, x_{P_i})$ where $\gamma $ is symmetric in $x_S$ , $|S| = \ell _i - 1$ , with $S \subseteq [2]$ , $c \in \{3, 4\}$ , or

• • $\gamma (x_S, x_c, x_{P_i})$ where $\gamma $ is symmetric in $x_S$ , $|S| = \ell _i - 1$ , with $c \in \{3, 4\}$ , $S \subseteq [4] \setminus \{c\}$ .

Consider all $\gamma $ above and replace variables $x_S$ by $z_1, \dots , z_{\ell _i}$ when $x_c$ does not appear, and $x_S$ by $z_1, \dots , z_{\ell _i - 1}$ and $x_c$ by $z_{\ell _i}$ when $x_c$ appears. Every form is thus symmetric in $z_{[\ell _i - 1]}$ . Let $C, D \geq 2$ quantities to be specified later, which will depend only on the constants in Lemma 3.1. Apply Lemma 4.1 to the list. We thus obtain integers $s,q,t$ such that $s + q + t \leq r$ and multilinear forms $\sigma _{[s]}, \pi _{[q]}, \rho _{[t]}$ with properties (i)–(v) from that lemma for a parameter $R \leq C^{D^{O(r)}}$ . We add superscript $\sigma ^{i, \ell _i}_1 \dots $ to stress the dependence on i and $\ell _i$ . Note that by Remark 4.2, when $\ell _i = 2$ , we only have forms $\sigma _{[s]}$ and $\rho _{[t]}$ , and when $\ell _i = 1$ , we only have forms $\sigma _{[s]}$ . Furthermore, since we are applying Lemma 4.1 to forms in at most three variables, low partition rank means that we may pass to a subspace where we get exact properties. Thus, by passing to a further subspace U, we may assume that forms $\sigma _{[s]}$ are symmetric in their active variables. To be precise, the subspace U has a codimension at most $3rR + r$ inside G (recall that we have first passed to a subspace of codimension at most r in order to remove the irrelevant partitions).

Note that the forms $\pi _i$ and $\rho _i$ have the property that they are symmetric in the first $\ell _i - 1$ variables, the variables $x_{P_i}$ play a passive role, and $z_{\ell _i}$ is related to $z_{[\ell _i-1]}$ in the sense that we shall consider compositions with transpositions $(a\,\,\ell _i)$ . With this in mind, we adopt the following notation. For a multilinear form $\phi (z_{[\ell _i]}, x_{P_i})$ , we separate its variables as $\phi (z_{[\ell _i - 1]}\,|\,z_{\ell _i}\,|\,x_{P_i})$ where the first group indicated the symmetric part, the second group has a single variable, and the third group consists of passive variables. We refer to the variable in the second group as the asymmetric variable.

Replace forms $\beta (x_I, x_{P_a})$ where $I \subseteq [4]$ , $|I| = \ell _a$ , by the newly found maps using property (v) of Lemma 4.1. If $3, 4 \notin I$ , then we know that $\beta $ is symmetric in $x_I$ so the second part of Observation 4.3 implies that $\beta (x_I, x_{P_a})$ can be replaced by

$$\begin{align*}\sum_{i \in [s]} \lambda_i \sigma_i(x_I, x_{P_a}) + \sum_{i \in [t], c' \in I} \lambda^{\prime\prime}_{i, c'} \rho_i(x_{I \setminus c'}\,|\,x_{c'}\,|\, x_{P_a}) + \tau(x_I, x_{P_a}),\end{align*}$$

where $\tau (x_I, x_{P_a})$ is a multilinear form of partition rank at most $(r+2)R$ . On the other hand, if $I \cap \{3, 4\} \not = \emptyset $ , take $c \in I\cap \{3, 4\}$ to be an index such that $\beta $ is symmetric in $x_{I \setminus \{c\}}$ . Properties (iii) and (v) of Lemma 4.1 and Observation 4.3 imply that $\beta (x_I, x_{P_a})$ can be replaced by

$$\begin{align*}\sum_{i \in [s]} \lambda_i \sigma_i(x_I, x_{P_a}) + \sum_{i \in [q]} \lambda^{\prime}_i \pi_i(x_{I \setminus c}\,|\,x_c\,|\, x_{P_a}) + \sum_{i \in [t], c' \in I} \lambda^{\prime\prime}_{i, c'} \rho_i(x_{I \setminus c'}\,|\,x_{c'}\,|\, x_{P_a}) + \tau(x_I, x_{P_a}),\end{align*}$$

where $\tau (x_I, x_{P_a})$ is a multilinear form of partition rank at most $(r + 2)R$ . Hence, forms $\pi _i$ always appear with $x_3$ or $x_{4}$ as the asymmetric variable.

In order to express $\alpha (x_{[5]})+ \alpha \circ (3\,\,4)(x_{[5]})$ in terms of these multilinear forms, we need to set up further notation. We use richer indices than just natural numbers, of the form $(i, j, \mathsf {S}, A)$ , $(i, j, \mathsf {PS}, A, v)$ , and $(i, j, \mathsf {AS}, A, v)$ , where $i \in [2]$ , j is a suitable index, $A \subseteq [4]$ of size $|A| = \ell _i$ , letters $\mathsf {S}$ , $\mathsf {PS}$ , and $\mathsf {AS}$ indicate the type of form we take, and $v \in A$ . Using such an index $\boldsymbol {\mathsf {i}}$ , we define form $\psi _{\boldsymbol {\mathsf {i}}}$ as a multilinear form in variables $x_{A \cup P_i}$ (which are ordered by the values of indices) and

$$\begin{align*}\psi_{\boldsymbol{\mathsf{i}}}(x_{A \cup P_i}) = \begin{cases} \sigma^{i, \ell_i}_j (x_A, x_{P_i}), &\text{ if }\boldsymbol{\mathsf{i}} = (i, j, \mathsf{S}, A),\\ \pi^{i, \ell_i}_j (x_{A \setminus v} \,|x_v\, |\, x_{P_i}), &\text{ if }\boldsymbol{\mathsf{i}} = (i,j, \mathsf{PS}, A, v),\\ \rho^{i, \ell_i}_j (x_{A \setminus v} \,|x_v\,|\,x_{P_i}), &\text{ if }\boldsymbol{\mathsf{i}} = (i,j, \mathsf{AS}, A, v). \end{cases}\end{align*}$$

Let $A(\boldsymbol {\mathsf {i}})$ be the set A in the rich index $\boldsymbol {\mathsf {i}}$ . Let $\mathcal {I}$ be the set of pairs $(\boldsymbol {\mathsf {i}}_1,\boldsymbol {\mathsf {i}}_2)$ such that:

• • $\boldsymbol {\mathsf {i}}_1 = (1, \dots )$ and $\boldsymbol {\mathsf {i}}_2 = (2, \dots )$ and

• • $A(\boldsymbol {\mathsf {i}}_1) \cup A(\boldsymbol {\mathsf {i}}_2)$ is a partition of $[4]$ .