Polynomial Identity Testing and Reconstruction for Depth-4 Powering Circuits of High Degree
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910031485599744 |
|---|---|
| author | Shpilka, Amir Tal, Yann |
| author_facet | Shpilka, Amir Tal, Yann |
| contents | We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth-$4$ arithmetic circuits of the form \[ Σ^{[r]}\!\wedge^{[d]}\!Σ^{[s]}\!Π^{[δ]}. \] This model generalizes Waring decompositions and diagonal circuits, and captures sums of powers of low-degree sparse polynomials. Specifically, each circuit computes a sum of $r$ terms, where each term is a $d$-th power of an $s$-sparse polynomial of degree $δ$. This model also includes algebraic representations that arise in tensor decomposition and moment-based learning tasks such as mixture models and subspace learning.
We give deterministic worst-case algorithms for PIT and reconstruction in this model. Our PIT construction applies when $d>r^2$ and yields explicit hitting sets of size $O(r^4 s^4 n^2 d δ^3)$. The reconstruction algorithm runs in time $\textrm{poly}(n,s,d)$ under the condition $d=Ω(r^4δ)$, and in particular it tolerates polynomially large top fan-in $r$ and bottom degree $δ$.
Both results hold over fields of characteristic zero and over fields of sufficiently large characteristic. These algorithms provide the first polynomial-time deterministic solutions for depth-$4$ powering circuits with unbounded top fan-in. In particular, the reconstruction result improves upon previous work which required non-degeneracy or average-case assumptions.
The PIT construction relies on the ABC theorem for function fields (Mason-Stothers theorem), which ensures linear independence of high-degree powers of sparse polynomials after a suitable projection. The reconstruction algorithm combines this with Wronskian-based differential operators, structural properties of their kernels, and a robust version of the Klivans-Spielman hitting set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial Identity Testing and Reconstruction for Depth-4 Powering Circuits of High Degree Shpilka, Amir Tal, Yann Computational Complexity We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth-$4$ arithmetic circuits of the form \[ Σ^{[r]}\!\wedge^{[d]}\!Σ^{[s]}\!Π^{[δ]}. \] This model generalizes Waring decompositions and diagonal circuits, and captures sums of powers of low-degree sparse polynomials. Specifically, each circuit computes a sum of $r$ terms, where each term is a $d$-th power of an $s$-sparse polynomial of degree $δ$. This model also includes algebraic representations that arise in tensor decomposition and moment-based learning tasks such as mixture models and subspace learning. We give deterministic worst-case algorithms for PIT and reconstruction in this model. Our PIT construction applies when $d>r^2$ and yields explicit hitting sets of size $O(r^4 s^4 n^2 d δ^3)$. The reconstruction algorithm runs in time $\textrm{poly}(n,s,d)$ under the condition $d=Ω(r^4δ)$, and in particular it tolerates polynomially large top fan-in $r$ and bottom degree $δ$. Both results hold over fields of characteristic zero and over fields of sufficiently large characteristic. These algorithms provide the first polynomial-time deterministic solutions for depth-$4$ powering circuits with unbounded top fan-in. In particular, the reconstruction result improves upon previous work which required non-degeneracy or average-case assumptions. The PIT construction relies on the ABC theorem for function fields (Mason-Stothers theorem), which ensures linear independence of high-degree powers of sparse polynomials after a suitable projection. The reconstruction algorithm combines this with Wronskian-based differential operators, structural properties of their kernels, and a robust version of the Klivans-Spielman hitting set. |
| title | Polynomial Identity Testing and Reconstruction for Depth-4 Powering Circuits of High Degree |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2602.20832 |