For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )

Then is the maximum of the polynomial degrees of the

Then the existence of an eigenvalue is equivalent to the ideal generated by ( the relations satisfied by ) being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero.

Then the interpolating polynomial is formed as above using the topmost entries in each column as coefficients.

Let V be an algebraic set defined as the set of the common zeros of an ideal I in a polynomial ring over a field K, and let A = R / I be the algebra of the polynomials over V. Then the dimension of V is:

Then, for some quotient polynomial Q ( x ) and remainder polynomial R ( x ) with degree ( R ) < degree ( D ),

Then all vanish for sufficiently large n. One can then show for some polynomial P over rational numbers.

P is a polynomial of degree n − r. Then we have the following:

For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divides the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable ( which is the case for every p but a finite number ) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.

Then for polynomial functions f we have the Newton series:

Then for all non-uniform probabilistic polynomial time algorithms that output and of increasing length k, the probability that and is a negligible function in k.

Then det ( J-xI ) is a polynomial in x of degree n. If n is odd, then it has a real root, z.

Assume now that the polynomial has a unitary root of multiplicity d. Then it can be rewritten as:

Then the k-jet of f at the point is defined to be the polynomial

To prove Nakayama's lemma from the Cayley – Hamilton theorem, assume that IM = M and take φ to be the identity on M. Then define a polynomial p ( x ) as above.

Then if and only if can be written in the form where is also a polynomial.

Let A = K be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings, see Hilbert's syzygy theorem.

Then the polynomial

Then the coordinates of the d points of intersection are algebraic functions of the Plücker coordinates of U, and by taking a symmetric function of the algebraic functions, a homogeneous polynomial known as the Chow form ( or Cayley form ) of Z is obtained.

Then, since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line.

Then, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.

Then it would be that by recursion ( using the axiom of countable choice ) that a sequence of polynomials could be found so that, letting of minimal degree.

Then, when similar diagram is plotted for a cluster whose distance is not known, the position of the main sequence can be compared to that of the first cluster and the distance estimated.

Then there does not exist a strictly increasing sequence of open sets ( equivalently

Then there is an exact sequence relating the kernels and cokernels of a, b, and c:

Then it does whatever operation ( or sequence of operations ) it was going to do.

Then its negation ¬ φ, together with the field axioms and the infinite sequence of sentences 1 + 1 ≠ 0, 1 + 1 + 1 ≠ 0, …, is not satisfiable ( because there is no field of characteristic 0 in which ¬ φ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0 ).

Then the sheaf axioms can be expressed as the exactness of the sequence

Then there is a long exact sequence of homotopy groups

Then, in one afternoon, they wrote his entire body of work: 17 poems, none longer than a page, and all intended to be read in sequence under the title The Darkening Ecliptic.

Then A, B and their intersection A ∩ B are homotopy equivalent to circles, so the nontrivial part of the sequence yields

Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take M large enough to make the initial segment of terms up to c < sub > N </ sub > average to at most ε / 2, while each term in the tail is bounded by ε / 2 so that the average is also.

Then continue the sequence, where each subsequent term is the sum of the previous n terms.

Then there is a constant C depending only on b < sub > 1 </ sub >, ..., b < sub > s </ sub >, such that sequence

Then once a relationship is established contexts can be removed from site in the reverse order they arrived in accordance with the stratigraphic excavation and the creation of a Harris matrix for the sequence being investigated.

Then, in August, Cuellar accomplished a sequence of 35 batters gotten out in a row.

Then, in the novel's most controversial sequence, she opens her dress and offers her breast to Lecter ; he accepts her offer and the two became lovers.

" Hughes went on to comment positively on the episode's debut of the high-definition opening sequence, and its introduction of Joyce Kinney as a replacement for news anchor Diane Simmons, who was killed off in " And Then There Were Fewer ".

Then the sequence of Y < sub > i </ sub > has exactly the probability distribution of tosses made with the second coin.

Then the sequence is pure exact if and only if C is flat.

Then she wakes up, showing that she dreamed this sequence at least.

Then follows in sequence: the dance of the hours of dawn, the hours of day, the hours of the night and the morning.