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gondolf
gondolf
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Suppose $m,n$ are positive integers.

$D$ denotes the set of $n\times n$ complex semidefinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are $n\times n$ Hermitians.

We are interested in the projection of $D$ onto $A_1,\cdots,A_m$ in the following way,

$$ S=\{(tr(A_1N),\cdots,tr(A_m N)):N\in D\} $$

It is clear $S$ is convex compact set. How to characterize its boundary. Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough? How many polynomials are needed?

In particular, for the case $m=4$, what do we know? Can we compute the boundary efficiently?

Suppose $m,n$ are positive integers.

$D$ denotes the set of $n\times n$ complex semidefinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are $n\times n$ Hermitians.

We are interested in the projection of $D$ onto $A_1,\cdots,A_m$ in the following way,

$$ S=\{(tr(A_1N),\cdots,tr(A_m N)):N\in D\} $$

It is clear $S$ is convex compact set. How to characterize its boundary. Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough? How many polynomials are needed?

Suppose $m,n$ are positive integers.

$D$ denotes the set of $n\times n$ complex semidefinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are $n\times n$ Hermitians.

We are interested in the projection of $D$ onto $A_1,\cdots,A_m$ in the following way,

$$ S=\{(tr(A_1N),\cdots,tr(A_m N)):N\in D\} $$

It is clear $S$ is convex compact set. How to characterize its boundary. Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough? How many polynomials are needed?

In particular, for the case $m=4$, what do we know? Can we compute the boundary efficiently?

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gondolf
gondolf
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On PolynopmialPolynomial Characterization of Projection area of semidefinite matrices

Suppose $m,n$ are positive integers.

$D$ denotedenotes the set of $n\times n$ complex semideinitesemidefinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are all $n\times n$ Hermitians.

We are interested in the projection of $D$ onto $A_1,\cdots,A_m$ in the following way,

$$ S=\{(tr(A_1N),\cdots,tr(A_m N)):N\in D\} $$

It is clear $S$ is convex compact set. How to charactrizecharacterize its boundary. Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough? How many polynomials are needed?

On Polynopmial Characterization of Projection area of semidefinite matrices

Suppose $m,n$ are positive integers.

$D$ denote the set of $n\times n$ complex semideinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are all $n\times n$ Hermitians.

We are interested in the projection of $D$ onto $A_1,\cdots,A_m$ in the following way,

$$ S=\{(tr(A_1N),\cdots,tr(A_m N)):N\in D\} $$

It is clear $S$ is convex compact set. How to charactrize its boundary. Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough?

On Polynomial Characterization of Projection area of semidefinite matrices

Suppose $m,n$ are positive integers.

$D$ denotes the set of $n\times n$ complex semidefinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are $n\times n$ Hermitians.

We are interested in the projection of $D$ onto $A_1,\cdots,A_m$ in the following way,

$$ S=\{(tr(A_1N),\cdots,tr(A_m N)):N\in D\} $$

It is clear $S$ is convex compact set. How to characterize its boundary. Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough? How many polynomials are needed?

Source Link
gondolf
gondolf
  • 1.5k
  • 9
  • 15

On Polynopmial Characterization of Projection area of semidefinite matrices

Suppose $m,n$ are positive integers.

$D$ denote the set of $n\times n$ complex semideinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are all $n\times n$ Hermitians.

We are interested in the projection of $D$ onto $A_1,\cdots,A_m$ in the following way,

$$ S=\{(tr(A_1N),\cdots,tr(A_m N)):N\in D\} $$

It is clear $S$ is convex compact set. How to charactrize its boundary. Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough?