Geodesic
On the multiplicative complexity of polynomials
Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 85-89

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We study the multiplicative (nonscalar) complexity $\mu(M_{d,n})$ for the class $M_{d,n}$ of complex polynomials in $n$ variables of degree at most $d$. It is shown that the relation $\mu(M_{d,n}) \asymp n^{\lceil d/2\rceil}$ holds for any constant $d \ge 2$. The lower bound for odd values of $d$ in this relation is new. For the case of cubic polynomials we prove more accurate bounds $n^2/18 \lesssim \mu(M_{3,n}) \lesssim n^2/4$.
Keywords: arithmetic circuits, multiplicative complexity, polynomials 
@article{DM_2022_34_3_a6,
     author = {I. S. Sergeev},
     title = {On the multiplicative complexity of polynomials},
     journal = {Diskretnaya Matematika},
     pages = {85--89},
     publisher = {mathdoc},
     volume = {34},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2022_34_3_a6/}
}
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I. S. Sergeev. On the multiplicative complexity of polynomials. Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 85-89. http://geodesic.mathdoc.fr/item/DM_2022_34_3_a6/