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35-463: Ars Magna may refer to: Ars Magna (Cardano book) , a 16th-century book on algebra Ars Magna (Llull book) , a 14th-century philosophical work Ars Magna Lucis et Umbrae , a 17th-century work on optics Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Ars Magna . If an internal link led you here, you may wish to change

70-761: A ) k s ( x ) {\displaystyle p(x)=(x-a)^{k}s(x)} . If k = 1 {\displaystyle k=1} , then a is called a simple root . If k ≥ 2 {\displaystyle k\geq 2} , then a {\displaystyle a} is called a multiple root . For instance, the polynomial p ( x ) = x 3 + 2 x 2 − 7 x + 4 {\displaystyle p(x)=x^{3}+2x^{2}-7x+4} has 1 and −4 as roots , and can be written as p ( x ) = ( x + 4 ) ( x − 1 ) 2 {\displaystyle p(x)=(x+4)(x-1)^{2}} . This means that 1

105-490: A ∈ F {\displaystyle a\in F} is a root of multiplicity k {\displaystyle k} of p ( x ) {\displaystyle p(x)} if there is a polynomial s ( x ) {\displaystyle s(x)} such that s ( a ) ≠ 0 {\displaystyle s(a)\neq 0} and p ( x ) = ( x −

140-411: A set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct". In prime factorization , the multiplicity of a prime factor is its p {\displaystyle p} -adic valuation . For example, the prime factorization of the integer 60 is the multiplicity of the prime factor 2 is 2 , while the multiplicity of each of

175-465: A function at the point x ∗ = ( 0 , 0 ) {\displaystyle \mathbf {x} _{*}=(0,0)} . The multiplicity is always finite if the solution is isolated, is perturbation invariant in the sense that a k {\displaystyle k} -fold solution becomes a cluster of solutions with a combined multiplicity k {\displaystyle k} under perturbation in complex spaces, and

210-675: A polynomial equation with multiple roots is x  = 12 x  + 16, of which −2 is a double root. Ars Magna also contains the first occurrence of complex numbers (chapter XXXVII). The problem mentioned by Cardano which leads to square roots of negative numbers is: find two numbers whose sum is equal to 10 and whose product is equal to 40. The answer is 5 + √ −15 and 5 − √ −15 . Cardano called this "sophistic," because he saw no physical meaning to it, but boldly wrote "nevertheless we will operate" and formally calculated that their product does indeed equal 40. Cardano then says that this answer

245-586: A system of m {\displaystyle m} equations of n {\displaystyle n} variables with a solution x ∗ {\displaystyle \mathbf {x} _{*}} where f {\displaystyle \mathbf {f} } is a mapping from R n {\displaystyle R^{n}} to R m {\displaystyle R^{m}} or from C n {\displaystyle C^{n}} to C m {\displaystyle C^{m}} . There

280-737: A two-year span (1543–1545). In 1535, Niccolò Fontana Tartaglia became famous for having solved cubics of the form x  +  ax  =  b (with a , b  > 0). However, he chose to keep his method secret. In 1539, Cardano, then a lecturer in mathematics at the Piatti Foundation in Milan, published his first mathematical book, Pratica Arithmeticæ et mensurandi singularis ( The Practice of Arithmetic and Simple Mensuration ). That same year, he asked Tartaglia to explain to him his method for solving cubic equations . After some reluctance, Tartaglia did so, but he asked Cardano not to share

315-476: A vector space, called the Macaulay dual space at x ∗ {\displaystyle x_{*}} , and its dimension is the multiplicity of x ∗ {\displaystyle x_{*}} as a zero of f {\displaystyle f} . Let f ( x ) = 0 {\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} } be

350-415: Is K [ X ] / ⟨ ( X − α ) m i ⟩ . {\displaystyle K[X]/\langle (X-\alpha )^{m_{i}}\rangle .} This is a vector space over K , which has the multiplicity m i {\displaystyle m_{i}} of the root as a dimension. This definition of intersection multiplicity, which

385-433: Is "as subtle as it is useless". It is a common misconception that Cardano introduced complex numbers in solving cubic equations. Since (in modern notation) Cardano's formula for a root of the polynomial x  +  px  +  q   is square roots of negative numbers appear naturally in this context. However, q /4 +  p /27 never happens to be negative in the specific cases in which Cardano applies

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420-401: Is a root of multiplicity k − 1 {\displaystyle k-1} of the derivative of that polynomial, unless the characteristic of the underlying field is a divisor of k , in which case a {\displaystyle a} is a root of multiplicity at least k {\displaystyle k} of the derivative. The discriminant of a polynomial

455-400: Is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of the fundamental theorem of algebra . If a {\displaystyle a} is a root of multiplicity k {\displaystyle k} of a polynomial, then it

490-413: Is also a Macaulay dual space of differential functionals at x ∗ {\displaystyle \mathbf {x} _{*}} in which every functional vanishes at f {\displaystyle \mathbf {f} } . The dimension of this Macaulay dual space is the multiplicity of the solution x ∗ {\displaystyle \mathbf {x} _{*}} to

525-641: Is an important Latin -language book on algebra written by Gerolamo Cardano . It was first published in 1545 under the title Artis Magnae, Sive de Regulis Algebraicis Liber Unus ( Book number one about The Great Art, or The Rules of Algebra ). There was a second edition in Cardano's lifetime, published in 1570. It is considered one of the three greatest scientific treatises of the early Renaissance , together with Copernicus ' De revolutionibus orbium coelestium and Vesalius ' De humani corporis fabrica . The first editions of these three books were published within

560-417: Is essentially due to Jean-Pierre Serre in his book Local Algebra , works only for the set theoretic components (also called isolated components ) of the intersection, not for the embedded components . Theories have been developed for handling the embedded case (see Intersection theory for details). Let z 0 be a root of a holomorphic function f , and let n be the least positive integer such that

595-471: Is everywhere non-negative if and only if all its roots have even multiplicity and there exists an x 0 {\displaystyle x_{0}} such that f ( x 0 ) > 0 {\displaystyle f(x_{0})>0} . For an equation f ( x ) = 0 {\displaystyle f(x)=0} with a single variable solution x ∗ {\displaystyle x_{*}} ,

630-514: Is identical to the intersection multiplicity on polynomial systems. In algebraic geometry , the intersection of two sub-varieties of an algebraic variety is a finite union of irreducible varieties . To each component of such an intersection is attached an intersection multiplicity . This notion is local in the sense that it may be defined by looking at what occurs in a neighborhood of any generic point of this component. It follows that without loss of generality, we may consider, in order to define

665-469: Is reduced to the single point P . Therefore, the local ring at this component of the coordinate ring of the intersection has only one prime ideal , and is therefore an Artinian ring . This ring is thus a finite dimensional vector space over the ground field. Its dimension is the intersection multiplicity of V 1 and V 2 at W . This definition allows us to state Bézout's theorem and its generalizations precisely. This definition generalizes

700-402: Is zero if and only if the polynomial has a multiple root. The graph of a polynomial function f touches the x -axis at the real roots of the polynomial. The graph is tangent to it at the multiple roots of f and not tangent at the simple roots. The graph crosses the x -axis at roots of odd multiplicity and does not cross it at roots of even multiplicity. A non-zero polynomial function

735-531: The Taylor expansions of g and h about a point z 0 , and find the first non-zero term in each (denote the order of the terms m and n respectively) then if m  =  n , then the point has non-zero value. If m > n , {\displaystyle m>n,} then the point is a zero of multiplicity m − n . {\displaystyle m-n.} If m < n {\displaystyle m<n} , then

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770-488: The n derivative of f evaluated at z 0 differs from zero. Then the power series of f about z 0 begins with the n term, and f is said to have a root of multiplicity (or “order”)  n . If n  = 1, the root is called a simple root. We can also define the multiplicity of the zeroes and poles of a meromorphic function . If we have a meromorphic function f = g h , {\textstyle f={\frac {g}{h}},} take

805-467: The coefficients of f . If f ( X ) = ∏ i = 1 k ( X − α i ) m i {\displaystyle f(X)=\prod _{i=1}^{k}(X-\alpha _{i})^{m_{i}}} is the factorization of f , then the local ring of R at the prime ideal ⟨ X − α i ⟩ {\displaystyle \langle X-\alpha _{i}\rangle }

840-404: The equation f ( x ) = 0 {\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} } . The Macaulay dual space forms the multiplicity structure of the system at the solution. For example, the solution x ∗ = ( 0 , 0 ) {\displaystyle \mathbf {x} _{*}=(0,0)} of the system of equations in

875-478: The fact that Scipione del Ferro had discovered Tartaglia's formula before Tartaglia himself, a discovery that prompted him to publish these results. The book, which is divided into forty chapters, contains the first published algebraic solution to cubic and quartic equations . Cardano acknowledges that Tartaglia gave him the formula for solving a type of cubic equations and that the same formula had been discovered by Scipione del Ferro. He also acknowledges that it

910-408: The form x  +  ax  +  bx  +  c  = 0 to cubic equations without a quadratic term, but, again, he has to consider several cases. In all, Cardano was driven to the study of thirteen different types of cubic equations (chapters XI–XXIII). In Ars Magna the concept of multiple root appears for the first time (chapter I). The first example that Cardano provides of

945-692: The form of f ( x ) = 0 {\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} } with is of multiplicity 3 because the Macaulay dual space is of dimension 3, where ∂ i j {\displaystyle \partial _{ij}} denotes the differential functional 1 i ! j ! ∂ i + j ∂ x 1 i ∂ x 2 j {\displaystyle {\frac {1}{i!j!}}{\frac {\partial ^{i+j}}{\partial x_{1}^{i}\,\partial x_{2}^{j}}}} applied on

980-443: The formula. Multiplicity (mathematics)#Multiplicity of a root of a polynomial The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever

1015-401: The information until he published it. Cardano submerged himself in mathematics during the next several years working on how to extend Tartaglia's formula to other types of cubics. Furthermore, his student Lodovico Ferrari found a way of solving quartic equations, but Ferrari's method depended upon Tartaglia's, since it involved the use of an auxiliary cubic equation. Then Cardano became aware of

1050-449: The intersection multiplicity, the intersection of two affines varieties (sub-varieties of an affine space). Thus, given two affine varieties V 1 and V 2 , consider an irreducible component W of the intersection of V 1 and V 2 . Let d be the dimension of W , and P be any generic point of W . The intersection of W with d hyperplanes in general position passing through P has an irreducible component that

1085-417: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Ars_Magna&oldid=1098834880 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Ars Magna (Cardano book) The Ars Magna ( The Great Art , 1545)

Ars Magna - Misplaced Pages Continue

1120-899: The multiplicity is k {\displaystyle k} if In other words, the differential functional ∂ j {\displaystyle \partial _{j}} , defined as the derivative 1 j ! d j d x j {\displaystyle {\frac {1}{j!}}{\frac {d^{j}}{dx^{j}}}} of a function at x ∗ {\displaystyle x_{*}} , vanishes at f {\displaystyle f} for j {\displaystyle j} up to k − 1 {\displaystyle k-1} . Those differential functionals ∂ 0 , ∂ 1 , ⋯ , ∂ k − 1 {\displaystyle \partial _{0},\partial _{1},\cdots ,\partial _{k-1}} span

1155-438: The multiplicity of a root of a polynomial in the following way. The roots of a polynomial f are points on the affine line , which are the components of the algebraic set defined by the polynomial. The coordinate ring of this affine set is R = K [ X ] / ⟨ f ⟩ , {\displaystyle R=K[X]/\langle f\rangle ,} where K is an algebraically closed field containing

1190-388: The prime factors 3 and 5 is 1 . Thus, 60 has four prime factors allowing for multiplicities, but only three distinct prime factors. Let F {\displaystyle F} be a field and p ( x ) {\displaystyle p(x)} be a polynomial in one variable with coefficients in F {\displaystyle F} . An element

1225-403: Was Ferrari who found a way of solving quartic equations. Since at the time negative numbers were not generally acknowledged, knowing how to solve cubics of the form x  +  ax  =  b did not mean knowing how to solve cubics of the form x  =  ax  +  b (with a , b  > 0), for instance. Besides, Cardano also explains how to reduce equations of

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