# Questions tagged [algebraic-complexity]

The algebraic-complexity tag has no usage guidance.

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### Can the Nullstellensatz over $\mathbb{C}^n$ be certified by repeatedly guessing witnesses in $\mathbb{Z}^n$?

In Hilbert's Nullstellensatz is in the Polynomial Hierarchy, P. Koiran showed that, given a system of $S$ of $m$ polynomials on $n$ variables of maximum degree $d$, along with a number $x_0$ ...

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### How quickly can we mutliply Cayley-Dickson hypercomplexes?

Assuming that all of the coordinates of two Cayley-Dickson Hypercomplex numbers are non-negative integers less than a prime $p$, how quickly can we multiply these numbers? I'm also interested in what ...

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### Bit complexity versus arithmetic complexity of polynomial multiplication

Given degree $d_1$ and $d_2$ polynomials in $\Bbb Z[x]$ with coefficient sizes of bits $b_1$ and $b_2$ respectively
(1) what is the bit complexity of multiplying the two polynomials?
(2) What is ...

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### $p$-Adic or arithmetic variants of Khovanskii's “low complexity $\Rightarrow$ tame topology” theory

This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real ...

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### Can we define a height function for a variety over a finite field?

That is, is there a way to measure the complexity of a point over a finite field the same way we do it over number fields?

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### What is the complexity of intersecting two matrix algebras over a finite field?

The following question arose in a joint project with Arkadius Kalka and Adi Ben-Zvi.
Let $\mathbb{F}$ be a finite field, and $M_n(\mathbb{F})$ be the $n\times n$ matrices over $\mathbb{F}$.
For a ...

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### How large do algebraic representations need to be for packing circles in squares?

(This question is inspired by Erich's Packing Center.
I'm just asking about circles in squares to keep things simple, since I suspect
any answer would apply just-as-well to the rest of the problems ...

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### Consequences failure of $\tau$ conjecture

A question Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$ was asked on constructing $ak!$ with ring operations.
$\tau$ conjecture states if $\exists$ ...

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### The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...

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### Gaussian Elimination in terms of Group Action

Gaussian elimination makes determinant of a matrix polynomial-time computable. The reduction of complexity in computing the determinant, which is otherwise sum of exponential terms, is due to presence ...

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### What is the largest tensor rank of $n \times n \times n$ tensor?

The tensor rank of a three dimensional array $M[i,j,k], i,j,k\in [1,\ldots,n]$ is the minimal number of vectors $x_i,y_i,z_i$, such that $M=\sum_{i=1}^d x_i\otimes y_i\otimes z_i$.
From dimension ...