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The propositions for the degree of sums and products of polynomials in the above section do not apply, if any of the polynomials involved is the zero polynomial. [ 8 ] It is convenient, however, to define the degree of the zero polynomial to be negative infinity , − ∞ , {\displaystyle -\infty ,} and to introduce the arithmetic rules [ 9 ]
Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial that ignores all its variables and always returns zero. The lemma states that evaluating a nonzero polynomial on inputs chosen randomly from a large-enough set is likely to find an input that produces a nonzero output.
Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). [10] The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero ...
However, these counterexamples rely on −1 having a square root. If we take a field where −1 has no square root, and every polynomial of degree n ∈ I has a root, where I is any fixed infinite set of odd numbers, then every polynomial f(x) of odd degree has a root (since (x 2 + 1) k f(x) has a root, where k is chosen so that deg(f) + 2k ∈ I).
In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined, [9] defined to be −1, [10] or defined to be a −∞. [11] A constant polynomial is either the zero polynomial, or a polynomial of degree zero.
If r is not zero, then r / c m (writing c m ∈ F for the non-zero coefficient of highest degree in r) is a monic polynomial of degree m < n such that r / c m ∈ J α (because the latter is closed under multiplication/division by non-zero elements of F), which contradicts our original assumption of minimality for n.
More formally: a polynomial is a function P from N (natural numbers) to R (where R is a ring) where there is n in N (the degree) so that P(n) is not zero and for all m>n P(m)=0. The thing is that the definition of the degree is embedded in the definition of the polynomial.
If its minimal polynomial has degree n, then the algebraic number is said to be of degree n. For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational. The algebraic numbers are dense in the reals. This follows from the fact they contain the rational numbers, which are dense in the reals ...