# Questions tagged [determinant]

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### Coefficients of a determinant of a matrix of univariate polynomials is in $GapL$

Given any matrix of univariate polynomials of degree $\leq n^{O(1)}$ then prove that the coefficent of $x^i$ in the determinant of the matrix is in $GapL$ Hint: Use Mahajan-Vinay's result of ...
56 views

### Is there a construction which multiplies and adds spanning trees in Logspace?

I.1 Suppose we have two planar graphs $G_1$ and $G_2$ with spanning tree count $C_1$ and $C_2$ respectively then is there a graph construction in Logspace to get a planar graph from $G_1$ and $G_2$ ...
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137 views

### On perm+1 and det+1

Given a balanced bipartite graph G and a planar graph H. We do not know the number of perfect matchings in G and we do not know the number of spanning trees in H. But assume they are at least 3 both. ...
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143 views

### Can a sum of polynomially many determinants be expressed as a single determinant of a poly-size matrix?

(copied from a mathoverflow question because I realized this may be more appropriate for it) Let $A_1,A_2,...,A_k$ be $N$-by-$N$ matrices, with indeterminate entries in some field (say real or complex ...
439 views

### How many multiplications are needed to compute the determinant of a 3×3 matrix?

In a comment on this question in 2016, Jeffrey Shallit remarked: I've asked experts about this, and apparently it is not even currently known whether or not 9 multiplications are needed to compute ...
189 views

### "Addition function" that works for both perm and det simultaneously?

For $f = (f_n)$ a family of polynomials where $f_n$ is a polynomial in $n^2$ variables (which we can think of as the entries of an $n \times n$ matrix), say a function $S(A,B)$ is an addition function ...
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### On Determinant modulo $2^k$ complexity

Determinant of integer matrix modulo $2$ is complete for the class $\oplus L$. Is determinant modulo $2^k$ computable in $\oplus L$ at any fixed $k$? How about if $k=o(n)$ where matrix is $n\times n$?
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149 views

### Is the following problem in $coNP$?

Given an $n\times n$ matrix $M$ with $\mathbb Z$ entries is 'does an $\frac n2\times\frac n2$ minor of $M$ vanish?' in $\bf{coNP}$? At least one $\frac n2\times\frac n2$ minor non-vanish implies rank ...
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156 views

### Application of weak determinantal identities to GCT?

In general determinants have many identities. Would it help the $GCT$ program by invoking the paradigm of identities such as to state that if the permanent is converted to determinant then it has to ...
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209 views

### Complexity of $\{0,\pm1\}$ determinant in sparse cases?

If $M\in\{-1,0,+1\}^{n\times n}$ be a matrix with only $O(n)$ non-zero entries and hadamard product $M\odot M$ being symmetric can we compute $Det(M)$ in $O(n)$ bit complexity? Assume that the matrix ...
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369 views

### Maximum Polyhedron Volume in Given $n$ Points

Suppose we are given $n$ points $v_1,v_2,\cdots, v_n\in \mathbb{R}^k$, I want to find $k+1$ points $v_{i_1}, v_{i_2},\cdots,v_{i_{k+1}}$ such that the volume of the convex body spanned by them ...
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### Complexity of computing generalised determinants. (P - #P transition)

Computing the determinant of a matrix can be done in polynomial time, while computing the permanent is known to be #P-hard. Let $A$ be an $n \times n$ matrix. Define a generalised determinant function ...
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### The minimum number of arithmetic operations to compute the determinant

Has there been any work on finding the minimum number of elementary arithmetic operations needed to compute the determinant of an $n$ by $n$ matrix for small and fixed $n$? For example, $n=5$.
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1 vote
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### A question about a claim in "No occurrence obstructions in geometric complexity theory"

It the new preprint Peter Bürgisser, Christian Ikenmeyer, Greta Panova, "No occurrence obstructions in geometric complexity theory", 2016 it is stated that 1.3. Conjecture (Mulmuley and Sohoni ...
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197 views

### Implications of a recent negative result to geometric complexity

A paper was posted in arxiv http://arxiv.org/pdf/1512.03798.pdf titled 'Rectangular Kronecker coefficients and plethysms in geometric complexity theory' by Christian Ikenmeyer and Greta Panova with ...
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250 views

### What is the status of Determinantal Complexity of Permanent

Recently Landsberg and Ressayre announced an exponential lower bound for permanent's determinantal complexity assuming certain symmetry conditions. What is the status of the problem of Permanent's ...
127 views

### Complexity of a particular determinant

Suppose we have an $n\times n$ matrix $A$ with non-negative integer entries such that $\mathsf{Tr}(A^i)=0$ at every $i\in\{1,2,\dots,n-2,n-1\}$ and $\mathsf{Tr}(A^n)\neq0$, then from Trace-Determinant ...
815 views

### Expressing Determinant as Permanent

One major problem in TCS is the problem of expressing a permanent as a determinant. I was reading Agrawal's paper Determinant Versus Permanent and in one paragraph he claims the reverse problem is ...
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### state-of-the-art bit complexity of the determinant

I'm trying to understand the full bit-complexity of computing the determinant of an $n\times n$ integer matrix, with each entry represented by $M$ bits. I would like to know what is the state-of-the-...
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### Determinants and Matrix Multiplication - Similarity and differences in algorithmic complexity and arithmetic circuit size

I am trying to understand the relation between algorithmic complexity and circuit complexity of Determinants and Matrix Multiplication. It is known that the determinant of an $n\times n$ matrix can ...
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### Cancellation and determinant

Berkowitz algorithm provides a polynomial size circuit with logarithmic depth for determinant of a square matrix using matrix powers. The algorithm implicitly uses cancellation. Is cancellation ...
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533 views

### Implications of approximating the determinant

It is known that one can compute exactly the determinant of an $n\times n$ matrix in determinstic $\log^2(n)$ space. What would be the complexity implications of approximating the determinant of a ...
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305 views

### Grigoriev-Karpinski for the permanent

Grigoriev and Karpinski (ps.Z) showed that any depth-3 circuit over a fixed finite field computing $\mathrm{Det}_n$ requires $2^{\Omega(n)}$ size. I had the misconception(?) until recently that the ...
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195 views

### Counting small terms in a determinant calculation over polynomials (counting spanning trees by weight)

I have a $n\times n$ matrix $A$. It's terms are $a_{ij}=-x^{w_{ij}}$ if $i\neq j$ and $a_{ii}=\sum_{j=0}^{n+1} x^{w_{ij}}$ on the diagonal. The matrix is symmetric as $w_{ij}=w_{ji}$. Numbers $w_{ij}$ ...
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### Lower bound for determinant and permanent

In light of the recent chasm at depth-3 result (which among other things yields a $2^{\sqrt{n}\log{n}}$ depth-3 arithmetic circuit for the $n \times n$ determinant over $\mathbb{C}$), I have the ...
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929 views

### Smallest known formula for the determinant

The smallest known formula for the determinant has size $n^{\mathcal O(\log n)}$ according to the folklore (or to Ran Raz in its paper Multi-Linear Formulas for Permanent and Determinant are of Super-...
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