# On a condition number of general random polynomial systems

@article{Nguyen2016OnAC,
title={On a condition number of general random polynomial systems},
author={Hoi H. Nguyen},
journal={Math. Comput.},
year={2016},
volume={85},
pages={737-757}
}
• H. Nguyen
• Published 24 June 2015
• Mathematics
• Math. Comput.
Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of probability distributions. Our work also extends to certain perturbed systems.

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## References

SHOWING 1-10 OF 27 REFERENCES

### Eigenvalues and condition numbers of random matrices

Given a random matrix, what condition number should be expected? This paper presents a proof that for real or complex $n \times n$ matrices with elements from a standard normal distribution, the ex...

### Complexity of Bezout's Theorem V: Polynomial Time

• Mathematics, Computer Science
Theor. Comput. Sci.
• 1994

### Invertibility of random matrices: norm of the inverse

Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A-1 does not exceed Cn3 / 2

### Small Ball Probability, Inverse Theorems, and Applications

• Mathematics
• 2013
Let ξ be a real random variable with mean zero and variance one and A ={a 1; …; a n } be a multi-set in R d . The random sum $$S_A : = a_1 \xi _1 + \cdots + a_n \xi _n$$ where ξ i are iid

### Smallest singular value of a random rectangular matrix

• Mathematics
• 2008
We prove an optimal estimate of the smallest singular value of a random sub‐Gaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed sub‐Gaussian

### Complexity of Bezout’s Theorem II Volumes and Probabilities

• Mathematics
• 1993
In this paper we study volume estimates in the space of systems of n homegeneous polynomial equations of fixed degrees d i with respect to a natural Hermitian structure on the space of such systems

### Smooth analysis of the condition number and the least singular value

• Mathematics
Math. Comput.
• 2010
A general estimate is given for the condition number and least singular value of the matrix M + N n, generalizing an earlier result of Spielman and Teng for the case when x is gaussian and involves the norm ∥M∥.

### Inverse Littlewood-Offord theorems and the condition number of random discrete matrices

• Mathematics
• 2005
Consider a random sum r)\V\ + • • • + r]nvn, where 771, . . . , rin are independently and identically distributed (i.i.d.) random signs and vi, . . . , vn are integers. The Littlewood-Offord problem

### A numerical algorithm for zero counting. II: Distance to ill-posedness and smoothed analysis

• Mathematics
ArXiv
• 2009
A Condition Number Theorem is shown that this condition number of zero counting for real polynomial systems equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros).