arXiv subjects

Erich L. Kaltofen

Publications and source records attributed to Erich L. Kaltofen.

4 recordsLinked to original sources

arXiv2022

A note on the van der Waerden conjecture on random polynomials with symmetric Galois group for function fields

Let f(x) = x^n + (a[n-1] t + b[n-1]) x^(n-1) + ... + (a[0] t + b[0]) be of constant degree n in x and degree <= 1 in t, where all a[i],b[i] are randomly and uniformly selected from a finite field GF(q) of q elements. Then the probability that the Galois group of f over the rational function field GF(q)(t) is the symmetric group S(n) on n elements is 1 - O(1/q). Furthermore, the probability that the Galois group of f(x) over GF(q)(t) is not S(n) is >= 1/q for n >= 3 and > 1/q - 1/(2q^2) for n = 2.

math.NT↗

arXiv2020

Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding

We present sparse interpolation algorithms for recovering a polynomial with $\le B$ terms from $N$ evaluations at distinct values for the variable when $\le E$ of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars $\mathsf{K}$ and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of $\mathsf{K}$ is $\ne 2$. Our algorithms return a list of valid sparse interpolants for the $N$ support points and run in polynomial-time. For standard power basis our algorithms sample at $N = \lfloor \frac{4}{3} E + 2 \rfloor B$ points, which are fewer points than $N = 2(E+1)B - 1$ given by Kaltofen and Pernet in 2014. For Chebyshev basis our algorithms sample at $N = \lfloor \frac{3}{2} E + 2 \rfloor B$ points, which are also fewer than the number of points required by the algorithm given by Arnold and Kaltofen in 2015, which has $N = 74 \lfloor \frac{E}{13} + 1 \rfloor$ for $B = 3$ and $E \ge 222$. Our method shows how to correct $2$ errors in a block of $4B$ points for standard basis and how to correct $1$ error in a block of $3B$ points for Chebyshev Basis.

cs.SC↗

arXiv2014

Sparse Polynomial Interpolation Codes and their decoding beyond half the minimal distance

We present algorithms performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. Based on the initial work by Comer, Kaltofen and Pernet [Proc. ISSAC 2012], we define the sparse polynomial interpolation codes and state that their minimal distance is precisely the length divided by twice the sparsity. At ISSAC 2012, we have given a decoding algorithm for as much as half the minimal distance and a list decoding algorithm up to the minimal distance. Our new polynomial-time list decoding algorithm uses sub-sequences of the received evaluations indexed by a linear progression, allowing the decoding for a larger radius, that is, more errors in the evaluations while returning a list of candidate sparse polynomials. We quantify this improvement for all typically small values of number of terms and number of errors, and provide a worst case asymptotic analysis of this improvement. For instance, for sparsity T = 5 with up to 10 errors we can list decode in polynomial-time from 74 values of the polynomial with unknown terms, whereas our earlier algorithm required 2T (E + 1) = 110 evaluations. We then propose two variations of these codes in characteristic zero, where appropriate choices of values for the variable yield a much larger minimal distance: the length minus twice the sparsity.

cs.SC↗

arXiv2012

Certificates of Impossibility of Hilbert-Artin Representations of a Given Degree for Definite Polynomials and Functions

We deploy numerical semidefinite programming and conversion to exact rational inequalities to certify that for a positive semidefinite input polynomial or rational function, any representation as a fraction of sums-of-squares of polynomials with real coefficients must contain polynomials in the denominator of degree no less than a given input lower bound. By Artin's solution to Hilbert's 17th problems, such representations always exist for some denominator degree. Our certificates of infeasibility are based on the generalization of Farkas's Lemma to semidefinite programming. The literature has many famous examples of impossibility of SOS representability including Motzkin's, Robinson's, Choi's and Lam's polynomials, and Reznick's lower degree bounds on uniform denominators, e.g., powers of the sum-of-squares of each variable. Our work on exact certificates for positive semidefiniteness allows for non-uniform denominators, which can have lower degree and are often easier to convert to exact identities. Here we demonstrate our algorithm by computing certificates of impossibilities for an arbitrary sum-of-squares denominator of degree 2 and 4 for some symmetric sextics in 4 and 5 variables, respectively. We can also certify impossibility of base polynomials in the denominator of restricted term structure, for instance as in Landau's reduction by one less variable.

math.OC↗