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TR24-026 | 15th February 2024 12:47
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#### A subquadratic upper bound on sum-of-squares compostion formulas

**Abstract:**
For every $n$, we construct a sum-of-squares identitity

\[ (\sum_{i=1}^n x_i^2) (\sum_{j=1}^n y_j^2)= \sum_{k=1}^s f_k^2\,,\]

where $f_k$ are bilinear forms with complex coefficients and $s= O(n^{1.62})$. Previously, such a construction was known with $s=O(n^2/\log n)$.

The same bound holds over any field of positive characteristic.