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The geometric description of linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
How general is this approach? The set of zeroes of homogeneous polynomials f(X1, X2, ..., Xk) of some degree in k variables, with coefficients in Fq, is a point set in PG(k − 1, q), an algebraic variety. Varieties are studied in algebraic geometry. We do not want to proceed in such generality, but it is interesting to note that, because of Theorem 17.1, the study of algebraic varieties is expected to have an impact on coding theory.
Two-Variable Weighted Shifts in Multivariable Operator Theory
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Since M(1) is invertible, the last three columns of the flat extension M(2) can be written in terms of the first three columns; that is, the columns labeled X2, YX and Y2 are linear combinations of 1, X and Y; moreover, e < c. Each of these column relations is associated with a quadratic polynomial in x and y, whose zero sets give rise to the so-called algebraic variety of Ω^3 [31]; concretely, V(Ω^3) ≔ ⋂p(X,Y)=0, degp≤2Z(p), where Z(p) denotes the zero set of p. In our case, the three column relations are X2=cXYX=fXY2=be(f−d)a(c−e)X+cd−efc−eY.
On the finiteness of accessibility test for nonlinear discrete-time systems
Published in International Journal of Control, 2021
Mohammad Amin Sarafrazi, Ewa Pawłuszewicz, Zbigniew Bartosiewicz, Ülle Kotta
Let us recall some basic facts from ideal theory. An idealI of a commutative ring is a subset of with the properties that if and then and . The radical of an ideal I of , denoted by , is the set . If I coincides with its own radical, then I is called a radical ideal. The real radical of I, denoted by , is the set of all for which there are natural m, k and such that . If I, J are ideals of , then (i) the real radical of I is an ideal of , (ii) , (iii) if then , see Bochnak, Coste, and Roy (1998). A semialgebraic (respectively semianalytic) set X is a set such that for every there is an open neighbourhood V of x with property that is a finite Boolean combination of sets and where are polynomial functions (respectively analytic functions). For a set , the Zariski closure of A is defined as the smallest algebraic variety containing A, and is denoted by .