{
  "$type": "site.standard.document",
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  "path": "/report/2026/084",
  "publishedAt": "2026-05-20T04:42:34.000Z",
  "site": "https://eccc.weizmann.ac.il",
  "textContent": "A depth-4 algebraic circuit with top fan-in $k$ and bottom fan-in $2$ is a circuit $\\Phi$ of the form $\\Phi = \\sum_{i=1}^k \\prod_{j=1}^{m_i} Q_{ij}$, where the polynomials $Q_{ij} \\in \\mathbb{K}[x_1, \\ldots, x_n]$ have degree at most $2$. The class of all such circuits is denoted by $\\Sigma^k \\Pi \\Sigma \\Pi^2$. We say that the circuit $\\Phi$ is an identity if it formally computes the zero polynomial. An important parameter of $\\Sigma^k \\Pi \\Sigma \\Pi^2$ circuits $\\Phi$ is their (linear) rank, which is defined as the vector space dimension of the polynomials $\\\\{Q_{ij}\\\\}_{i \\in [k], j \\in [m_i]}$. We prove that, when the base field $\\mathbb{K}$ is of characteristic zero, the rank of any (simple and minimal) $\\Sigma^k \\Pi \\Sigma \\Pi^2$ identity is upper bounded by a function which depends only on the top fan-in $k$. This result makes progress on [BMS13, Conjecture 28], being the first work to establish a bound on the rank of such identities that depends only on the top fan-in. Moreover, when combined with [BMS13, Theorem 2], our main result yields the first deterministic, \\emph{polynomial time} PIT algorithm for $\\Sigma^k \\Pi \\Sigma \\Pi^2$ circuits. One of the key components of our proof of the rank bounds is the derivation of an approximate Hansen-type result, which is interesting in its own right. This result can be seen as an algebraic and higher-dimensional analogue of the approximate Sylvester-Gallai result of [ADSW14], and a distinct approximate fractional Sylvester-Gallai result than the one from [GOPS23]. Additionally, we prove a robust version of it, in the spirit of the generalization of Hansen's theorem by [BDWY13].",
  "title": "TR26-084 |  Rank bounds and polynomial-time PIT for $\\Sigma^k \\Pi \\Sigma \\Pi^2$ circuits | \n\n\tAbhibhav Garg, \n\n\tRafael Mendes de Oliveira, \n\n\tAkash Sengupta, \n\n\tNir Shalmon, \n\n\tAmir Shpilka"
}