SSignatures of monic polynomials.
Norbert A’CampoFebruary 21, 2017
Abstract
Let P : C → C be a monic polynomial map of degree d ≥
1. Wecall the inverse image of the union of the real and imaginary axis thegeometric picture of the polynomial P . The geometric picture of amonic polynomial is a piece-wise smooth planar graph. Smooth iso-topy classes relative to the 4 d asymptotic ends at infinity of geometricpictures are called signatures. The set of signatures Σ d of monic de-gree d polynomials is finite. We give a combinatorial characterizationof the set of signatures Σ d and prove that the space of monic polyno-mials of given signature is contractible. This construction leads to areal semi-algebraic cell decompositionPol d = (cid:91) σ ∈ Σ d { P | σ ( P ) = σ } of the space Pol d of monic polynomials of degree d . In this cell de-composition the classical discriminant locus ∆ d appears as a union ofcells. The complement of the classical discriminant B d := Pol d \ ∆ d is a union of cells. The face operators of this cell decomposition ofthe space B d are explicitly given. Since B d is a classifying space forthe braid group, we obtain a finite complex that computes the groupcohomology of the braid group with integral coefficients. Let P : C → C be a polynomial mapping. We assume that P is monic, i.e.with leading coefficient 1. We call a polynomial P balanced if its sub-leadingcoefficient vanishes which says that the sum of its roots P − (0) weighted bymultiplicity equals 0. A unique Tschirnhausen substitution z = z − t willtransform a monic polynomial to a monic and balanced one. We call the1 a r X i v : . [ m a t h . AG ] F e b nverse image by the map P of the union of the real and the imaginary axisthe geometric picture π P of a monic polynomial P .Geometric pictures of monic polynomials are special graphs in the Gaus-sian plane C . Their combinatorial restrictions are listed in the followingstatement. Theorem 1.1
Let P ( z ) be a monic polynomial of degree d > . Its geometricpicture π P is a smooth graph in C with the following properties:1. The graph has no cycles. The graph is a forest. The non-compactedges are properly embedded in C .2. The complementary regions have a -colouring by symbols A, B, C, D .3. The edges are oriented smooth curves and have a -colouring by sym-bols R, I . They carry the symbol R if the edge separates D and A or B and C coloured regions. They carry the symbol I if the edge separates A and B or C and D coloured regions. The orientation is right-handed if one crossesthe edge from D to A or A to B , and left-handed if one crosses B to C or C to D .4. The picture has d edges that near infinity are asymptotic to the rays re kπi/ d , r > , k = 0 , , · · · , d − . The colours R, I alternate and theorientations of the R coloured and also the I coloured alternate between out-going and in-going.5. Near infinity the sectors are coloured in the counter-clockwise orienta-tion by the -periodic sequence of symbols A, B, C, D, A, B, · · · .6. The graph can have types of vertices: for the first types only A, B or B, C or C, D or D, A regions are incident and only edges of one color are in-cident, moreover for the fifth type, regions of all colours are incident and thecolours appear in the counter clock-wise orientation as A, B, C, D, A, B, · · · .So, in particular the graph has no terminal vertices.7. At all points p ∈ π P the germ of the graph π P is smoothly diffeomorphicto the germ at ∈ C of { z ∈ C | Re z k = 0 } for some k = 1 , , · · · . Proof
The real and imaginary axis decompose the complex plane C in fourregions coloured by A, B, C, D according to the signs of the real part and theimaginary part respectively ++ , + − , −− , − +. The real and the imaginaryaxis are coloured by R, I and are oriented by the gradients of the real and theimaginary part. In fact, this is the colouring and the orientation of the picture π z of the degree 1 polynomial z . The picture π P inherites the colouring forits regions and the colouring together with orientation of its edges by pullingback via the map P the colouring and orientation of π z . Properties 2 , , · · · Re ( P ) ∗ Im ( P ) : C → R is harmonic. Indeed, Re ( P ) ∗ Im ( P ) = 12 Im ( P ) and the imaginary part ofthe holomorphic map P is harmonic. A minimal cycle Z in π P is a simplyclosed curve and would bound an open bounded region U . Since the function Re ( P ) ∗ Im ( P ) vanishes along Z , we would have Re ( P ) ∗ Im ( P ) = 0 on U .It follows that the image P ( U ) of P = Re ( P ) + i Im ( P ) is contained in theunion of the real and the imaginary axis in C , contradicting the openness ofthe non constant holomorphic mapping P . Theorem 1.2
For given degree d , there exist only finitely many isotopyclasses of graphs satisfying the properties of Theorem . . Proof
We compactify the graph by adding 4 d ideal vertices at infinity, onefor each ray, see Property 4. Let v be the number of vertices and v (cid:48) be thenumber of inner (non-ideal) vertices of the compactified graph. Since thedegree of an inner vertex is at least 4, the number of incidence pairs of avertex, finite or ideal, and edge is at least 4 v (cid:48) + 4 d . So the number e of edgesis at least 2 v (cid:48) + 2 d . Since the graph is an non-empty forest, its Euler numberis at least 1. Hence 1 ≤ v − e ≤ ( v (cid:48) + 4 d ) − (2 v (cid:48) + 2 d )showing v (cid:48) ≤ d − v ≤ d −
1. The statement follows, since the numberof isotopy classes, relative to infinity, of planar forests with 4 d ideal fixedterminal vertices at infinity and at most 2 d − Definition
A signature of degree d is a smooth isotopy class of graphs thatsatisfy the 7 properties of Theorem 1 . C , asignature becomes a topological space of planar graphs. Classical theoremsof Rheinhold Baer, David Epstein and Jean Cerf in planar topology tell usthat a signature is a contractible space. See the thesis of Yves Ladegailleriefor the study of spaces of graphs in surfaces.The following theorems are the main results. Every signature is realizedby a monic polynomial. Theorem 1.3
Let σ be a signature of degree d > . Then there exists some P ∈ Pol d whose geometric picture belongs to σ . The space of monic polynomials with given signature is contractible.
Theorem 1.4
Let σ be a signature of degree d > . The space { P ∈ P ol d | σ ( P ) = σ } is contractible. Bi-regular polynomials.
As intermezzo we first study most generic monic polynomials. The corre-sponding cells are the open cells. We call the map P bi-regular if 0 ∈ R is a regular value for both mappings, the real aswell as the imaginary part Re ( P ) : C → R and Im ( P ) : C → R . The geometric picture of a bi-regularpolynomial P of degree d > ∈ R for the maps Re ( P ) and Im ( P ). It has d vertices of valency 4 at theroots of P and 4 d non compact terminal edges. Here we show as an examplethe geometric picture of the bi-regular polynomial P = z − z + z − z + 5 z + z + 3 + 2 i See the picture in Fig. 1 that we have made with SAGE. The blue linesare the inverse image by P of the oriented (from −∞ to + ∞ ) real axis,and are also the inverse image by Im ( P ) of 0 ∈ R . The green ones arethe inverse image by P of the oriented (from − i ∞ to + i ∞ ) imaginary axisand also the inverse image by Re ( P ) of 0 ∈ R . There are 13 transversalintersection points of a green and blue line, which of course are the roots ofthe polynomial P . At each root a blue and a green line intersect orthogonallysince the polynomial map P is conformal and hence its differential at regularpoints preserves angles. Each blue or green line is an properly embedded copyof the oriented real line in the plane. Near infinity those lines are asymptoticto rays emanating from the origin. For a bi-regular polynomial of degree d ,the inverse image of the real axis is a disjoint union of d copies of an orientedreal line, having 2 d ends that are asymptotic to rays directed by the 2 d -rootsof unity θ with θ d = ±
1. The inverse image of the imaginary axis is a similardisjoint union of d copies of a real line, except that the ends are asymptoticto the directions of the 4 d -roots of unity θ with θ d = ± i . We orient theasymptotic rays from 0 to ∞ . The orientation of a curve of the picture andits asymptotic ray match if θ d = +1 , + i and are opposite if θ d = − , − i . Wecall the geometric picture of a bi-regular polynomial a bi-regular picture.We say that two bi-regular pictures π, π (cid:48) are combinatorially equivalentif there exists a regular proper ambient isotopy that keeps the direction ofthe asymptotics fixed and that moves π to π (cid:48) . A bi-regular signature is acombinatorial equivalence class of bi-regular pictures.In the next section we will count the number of bi-regular signatures ofdegree d . A bi-regular signature is a signature such that every vertex hasvalence 4 at which the incident 4 sectors have 4 different colours.This is especially interesting for following a root r t continuously givenby a family P t of polynomials. What is still missing, is an understandingof the wall crossings phenomena between different connected components of4i-regular polynomials. In particular we do in particular not know the dualgraph of those components for which a component becomes a vertex and apair of vertices is connected by an edge if one gets from one component tothe other by a transversal wall crossing. We plan applications to computergraphics and robotics in future. -2 -1 0 1 2-2-1012 Fig. 1. P = z − z + z − z + 5 z + z + 3 + 2 i . Let P : C → C be a monic polynomial mapping of degree d . Weassume that 0 ∈ R is a regular value of the imaginary part mapping of Im ( P ) : C → R . The inverse image P − ( R ) ⊂ C is a system of d smoothlyembedded copies of the real line. The orientations of this systeme can bereconstructed, since the positive end of the real axis is an asymptotic raywith matching orientations and since the matching and non-matching endsalternate if one goes from one 2 d -root of unity to the next. So we can forgetthe orientations of the components of P − ( R ) ⊂ C without loosing informa-5ion. Combinatorially we can think of P − ( R ) ⊂ C as a system of d disjointdiagonals and edges in a 2 d -gon. The number of possible systems D ( d ) of d non intersecting diagonals or edges in a 2 d - gon is given by a Catalan num-ber. We put D (0) = 1, and have D (1) = 1 , D (2) = 2 and for d ≥ d nonintersecting diagonals or edges in a 2 d - gon is given by a Catalan number.Here in particular Disjoint means having no common vertices! Moreover, for d ≥ D ( d + 1) = (cid:88) ≤ i ≤ d D ( i ) D ( d − i )holds. This recurrence relation is obtained by splitting a 2( d + 1)-gon alongthe curve that has the first vertex as end. This is the recurrence relation forthe Catalan numbers, hence D ( d ) = d +1 (cid:0) dd (cid:1) . The first Catalan numbers for d = 1 , , · · · are 1 , , , , , , , D ( d ) possibilities.The two possibilities are very depending, since each component of the in-verse image of the real axis intersects transversally the inverse image of theimaginary axis once. So we need a combined counting. Let Pict( d ) be thenumber of possible combinatorial types of pictures. We put Pict(0) = 1 andhave Pict(1) = 1. For d ≥ d + 1) = (cid:88) ≤ i, ≤ j, ≤ k, ≤ l,i + j + k + l = d Pict( i )Pict( j )Pict( k )Pict( l )which is obtained from the following splitting: let A be the curve in P − ( R )that is asymptotic to the positive real axis and B be the curve in P − ( i R )that intersects A . The pair of curves ( A, B ) splits the complex plane in fourregions. The summing indices i, j, k, l are the number of roots of a bi-regularpolynomial in these regions. The Catalan recurrence expresses D ( d + 1) asa sum of products D ( a ) D ( b ) with a + b = d . The recurrence for Pict( d + 1)is similar, except that Pict( d + 1) is a sum of 4-factor products. In order tointegrate this recursion we first computed with a PARI program the first 15terms. The result was1 , , , , , , , , , , , , , , https:/oeis.org/ identifies this sequence with the sequence6 d ) = 13 d + 1 (cid:18) dd (cid:19) By induction upon d we check that the proposed expression satisfies therecursion relation of Pict( d ).The space of bi-regular polynomials of degree d is an open subset in thespace of all degree d polynomials. -3 -2 -1 0 1 2 3-3-2-10123 Fig. 2. P ( z ) = ( z − i/ + ( z − i/ + 1.The connected components correspond bijectively to pictures of bi-regularpolynomials. We say that two components are neighbours if separated by awall of real co-dimension 1. In this case we also say that two bi-regularpictures or bi-regular signaturs are neighbours. The signature of the pictureof Fig. 2 defines such wall that separates two bi-regular components. Thepicture of Fig. 2 allows two smoothings that yield bi-regular pictures.The discriminant ∆ ⊂ C d is the space of monic polynomial mappings P : C → C having 0 ∈ C as critical value. Clearly, a polynomial P belongs7o ∆ if and only if the mappings Re ( P ) : C → R and Im ( P ) : C → R havea critical point with critical value 0 in common. It follows that each cell ofbi-regular polynomials is contained in the complement of ∆. It also followsthat the co-dimension 1 walls are contained in the complement of ∆.The polynomial P ( z ) = z − z/ /
27 is regular above 0 as mappingfrom C to C , but the map Re ( P ) : C → R has two critical points with 0 asvalue. The polynomial P belongs to a stratum of real co-dimension 2. SeeFig. 3. The two critical points of Re ( P ) : C → R can fuse together in astratum of real co-dimension 3. See the picture of Q ( z ) = z + 1 in Fig. 4. -2 -1 0 1 2-2-1012 Fig. 3. P ( z ) = z − z + 1.The polynomial P ( z ) = z − z + 1 is regular above 0 as mapping from C to C , but the map Re ( P ) : C → R has two critical points with 0 as value.The polynomial P belongs to a stratum of real co-dimension 2. See Fig. 3.The two critical points of Re ( P ) : C → R can fuse together in a stratum ofreal co-dimension 3. See the picture of Q ( z ) = z + 1 in Fig. 4.The two critical points of Re ( P ) : C → R can be smoothed, one a lot,the other less, see Fig. 5. 8gain with a PARI program we could compute the numbers of co-dimension1 walls in degree d = 1 , , , ... . We get:0 , , , , , , , , , Problems.
Let B ( d, c ) be the number of cells in B d of codimension c . Studythe generating series C ( x, y ) = (cid:88) d,c B ( d, c ) x d y c ∈ Z [[ x, y ]]and the coefficients C c ( x ) = (cid:88) d B ( d, c ) x d ∈ Z [[ x ]]Study the differential operators that annilate C ( x, y ) , C c ( x ). Find closedexpressions for B ( d, c ). -2 -1 0 1 2-2-1012 Fig. 4. Q ( z ) = z + 1.9 Fig. 5. P ( z ) = z − ( 110 + i
200 ) z + 1 + i -3 -2 -1 0 1 2 3-3-2-10123 Fig. 6. T (5 , z ) = 16 z − z + 5 z .Special polynomials have typical pictures. As example see the fifth Cheby-shev polynomial of the first kind in Fig. 6. One observes that its picture can10e smoothed at 4 places. So, the fifth Chebyshev polynomial belongs to acell of codimension 4. This cell is in the closure of 2 cells of bi-regular poly-nomials. This holds for all degrees: the Chebyshev polynomial T n of degree n belongs to a cell of codimension n − n − bi-regular cellsmeet. Incidently, observe that 2 n − is the leading coefficient of the polyno-mial T n . The cell of the signature σ ( T n ) is the space of all real monic Morsedeformations of the polynomial z n with n − The proofs are based on the Riemann Mapping Theorem in combinationwith theorems of Reinhold Baer, David Epstein and Jean Cerf on homotopyversus isotopy and theorems of C.J. Earle and J. Eells on contractability ofconnected components of groups of diffeomorphism in dimension two.
Proof of Theorem 1.3.
Let σ be a signature and let γ be a smoothoriented, coloured embedded graph in the class σ . Let 4 d be the numberof ideal vertices. The 7 properties allow to construct a smooth function f : C → C such that the following holds.1. The graph γ is the inverse image by f of the union of the real and theimaginary axis.2. The map f is open with at most d − Df is positive at all regular points of f . At each criticalpoint of f the the germ of f is smoothly equivalent to the germ of z ∈ C (cid:55)→ z k + t ∈ C for some k = 1 , , · · · and some t ∈ { +1 , − , + i, − i, } .3. The restriction of f to an edge of γ is regular and injective.4. The colourings of regions and edges of γ are the pull-backs by f of thethe colourings of P z .5. lim z ∈ C , | z |→ + ∞ f ( z ) z d = 1.Let J be the pull-back by f of the standard conformal structure J on C to C . The map f : ( C , J ) → ( C , J ) is holomorphic. By the Riemannmapping theorem a biholomorphic map ρ : ( C , J ) → ( C , J ) exists. Indeed,by property 5 for f , the map extends to a self-map of C ∪ {∞} . Replacing ρ finally by a positive real multiple λρ the composition f ◦ ρ : C → C byRouch´e’s Theorem will be a monic polymial having a picture in the class of γ . Proof Theorem 1.4.
Consider the signature σ as a space Γ of smoothoriented planar graphs. The space Γ, if equipped with the topology inducedby the topology of the oriented arc-length parametrizations of the edges, is11ontractible by Theorems of Baer and Epstein. Given γ ∈ Γ, the space E γ with the smooth topology of functions f : C → C satisfying the 5 propertiesas in the previous Theorem is contractible. The space E Γ of pairs ( f, γ ) with γ ∈ Γ and f ∈ E γ by a Theorem of J. Cerf is the total space of a fiber bundle π : E Γ → Γ , ( f, γ ) (cid:55)→ γ . It follows that the space E Γ is contractible.The group G C , ∞ of orientation preserving diffeomorphisms of C extendingto C ∪ {∞} as a diffeomorphism with the identity as differential at ∞ , iscontractible. The group G C , ∞ acts with closed orbits and without fixed pointson E Γ . So the space E Γ /G C , ∞ is contractible. By the Riemann mappingTheorem, there exists in every G C , ∞ -orbit a unique pair ( f, γ ) such thatthe pull back by f of the standard conformal structure on C is again thestandard structure. In order to achieve uniqueness of the pair ( f, γ ), werequire moreover that f , now by Rouch´e’s Theorem a monic polynomial, isbalanced. It follows that the space of monic and balanced polynomials withpicture in Γ is a space of representatives for the quotient E Γ /G C , ∞ .We conclude that the space of monic balanced polynomial mappings P with picture in the isotopy class Γ is contractible. The group of Tschirn-hausen substitutions is contractible and acts fixed point free on the space ofmonic polynomial mappings P with picture in the isotopy class Γ. Hence,the space of monic polynomial mappings P with signature σ is contractibletoo. Labelling roots.
Let r be a root of a monic polynomial P . The root r belongs to a connected component T of the picture of P . The component T is an coloured oriented tree. We define as label the pair of ( α, β ) consisting ofthe 4 d -root of unity. The root of unity α = e πik / d , with k ∈ { , , · · · , d − } minimal, is in fact the 2 d -root of unity, that we get by starting at r and by following in T the oriented edges in P − ( R ). The root of unity β = e πi (2 k +1) / d with minimal k ∈ { , , · · · , d − } is the root of unity thatwe get by starting at r and by following in T the oriented edges in P − ( i R ).Essentially, from its label we can find back the corresponding root bysolving differential equations. The map from root to label is constant ineach cell of the cell decomposition by signatures. This property has clearlyapplications, each time one wishes to follow roots of polynomials continuouslyin families of polynomials. Robotics typically encounters this wish. The real axis R ∪ {∞} and the imaginary axis i R ∪ {∞} , both extended bythe point ∞ , divide the Riemann sphere P ( C ) = C ∪ {∞} in four regions,12hat again we label by the colours A, B, C, D . Define the picture of a rationalmap f : P ( C ) → P ( C ) be the inverse image of the union of the extendedaxis. Similarly, define the picture of a holomorphic map f : S → P ( C )on a Riemann surface. Call a rational map or more general a meromorphicfunction f on a Riemann surface very bi-regular if the critical values donot belong to the extended real or the extended imaginary axis. Call afunction f bi-regular if its critical values do not belong to the (non extended)real or imaginary axis. The bi-regular polynomials remain according to thisdefinition bi-regular. We plan to study in future from the combinatorial viewpoint these more general settings. The most non-bi-regular polynomial are P = ( z − r ) d , d ≥ , r ∈ C . Eachconnected component of bi-regular polynomials of degree d > P be bi-regular of degree d > t d P ( z/t ) , t ∈ R , t > , has as limit at t = 0 the polynomial z d . The family is an orbit of the weighted homogeneousaction of the group of positive real numbers( t, P ) (cid:55)→ t • P = t degree( P ) P ( z/t )In fact the larger group of affine substitutions z (cid:55)→ z/t + a, t > , a ∈ C acts on monic polynomials. We can use this action for simplification andnormalization. We define as norm (cid:107) P (cid:107) the l -norm of the vector of coefficientsof P . First, for a polynomial P (cid:54) = ( z − r ) d of degree d > z (cid:55)→ z − a killing the coefficient of the term z d − in P , next we choose t > P (cid:54) = z d there exists precisely one t > (cid:107) t • P (cid:107) = 1.So instead of studying the chambers of bi-regular polynomials in the vectorspace of complex dimension d of all monic polynomials of degree d , one canrestrict this study to the unit sphere of dimension 2 d − d Tchirnhausen simplified polynomials. For instance, it would be interestingto study what happens for d = 3. One gets a decomposition of the sphere S in 22 contractible components. What is the dual graph?Face operators correspond to the following two operations on signatures.Let σ be a signature. Let π be a picture in the class σ . The picture π decomposes the plane C in polygonal regions. The regions have piece-wisesmooth curves as boundaries. 13he first operation consists in contracting diagonals of regions. Let D bea smooth generic diagonal in such a region connecting two boundary points,such that the graph π ∪ D is still a forest. The endpoints of D are smoothpoints of edges. The new graph π ∪ D obtained by adding D to the picture π does not satisfy the 7 properties. In particular, two vertices are of degree3. Let π D be the planar graph obtained by contracting the diagonal D to anew vertex of degree 4. The graph π D , together with its colouring of edges,labelling of regions, satisfies the 7 properties and the class of π D is again asignature σ D .The second operation consists of contracting an edge of π connecting twovertices which are no roots, i.e. vertices not incident with all four colors A, B, C, D . Contracting the edge E in π transforms π to a new picture withthe 7 properties, so constructs a new signature σ E .The operation of adding and contracting a diagonal D to σ or the op-eration of contracting an edge E such that σ D or σ E is again a signaturecorresponds to a co-dimension one face operation for the cell decompositionof the space B d .It is a challenging problem to understand the combinatorics of these faceoperations and to describe the corresponding cell and co-chain complex forthe spaces B d .The semi-algebraic cell decomposition of B d is compatible with a tri-angulation by a theorem of Stanis l aw (cid:32)Lojasiewicz. Working in the secondbarycentric subdivisions allows to construct regular open neigborhoods U σ in B d of closures in B d of cells { P | σ ( P ) = σ } ⊂ B d . The integral ˇ C ech-cohomology of the acyclic covering { U σ } of B d computes the group cohomol-ogy H ∗ (( Br ( d ) , Z ) of the braid group Br ( d ). It is a challenging problem todo this computation effectivily.A third face operation is needed in spaces of polynomials that have rootsof multiplicities exceeding one. Instead of contracting edges, now also con-tracting minimal subtrees T with two or more edges in π such that the class σ T of π/T is again a signature gives a face operation.The unit sphere S d − in the space of complex Tchirnhausen simplifiedpolynomials of degree d has a natural probability measure. A natural ques-tion is about the probability that a random polynomial P realizes a givenpicture? Which picture has highest probability?The notion bi-regularity suggests two notions of discriminants for poly-nomials. We define as real discriminant the set ∆ R of all monic polynomials P of a given degree such that 0 ∈ R is a critical value of the real part Re ( P ),and accordingly ∆ i R all polynomials with 0 ∈ R as critical value for Im ( P ).Recall that the classical discriminant ∆ is the set of polynomials P suchthat the mapping P : C → C has 0 ∈ C as critical value. The complement14f the union ∆ R ∪ ∆ i R is the set of bi-regular polynomials and the classicaldiscriminant ∆ is included in the intersection ∆ R ∩ ∆ i R .We get a braid invariant as follows. A braid b defines an isotopy class ofa closed path of monic polynomials in the complement of ∆. What is theminimal number of bi-regular chambers that such a path for a given braidhas to visit? Acknowledgement.
This work started in 2014 at the Graduate School ofSciences, Hiroshima University during the Conference ”Branched Coverings,Degenerations and Related Topics”. The author thanks Professors MakotoSakuma and Ichiro Shimada for the warm hospitality and for providing stim-ulating mathematical environment. ” H e l l o P a r i ” Computes t h e v e c t o r o f t h e number o f p o s s i b l ep i c t u r e s o f t h e monic d e g r e e deg < = g b i − r e g u l a r p o l y n o m i a l s .v e c t o r n u m b p i c t ( g)= { X=v e c t o r ( g , i , 0 ) ;X[ 1 ] = 1 ;f o r ( deg =2 ,g ,f o r ( a =0 , deg − − − a ,f o r ( c =0 , deg − − a − b ,f o r ( d=deg − − a − b − c , deg − − a − b − c ,X [ deg ]=X [ deg ]+i f ( a ==0 ,1 ,X [ a ] ) ∗ i f ( b==0 ,1 ,X [ b ] ) ∗ i f ( c ==0 ,1 ,X [ c ] ) ∗ i d ( d==0 ,1 ,X [ d ] ) ;) ; ) ; ) ; ) ;) ;X } n u m b p i c t ( deg )= { X=v e c t o r n u m b p i c t ( deg ) ;X [ deg ] } { i f ( deg < − ∗ − ∗ deg ∗ sum ( a =1 , deg − ∗ X [ deg − a ] ) ;) ;Res } ” H e l l o Sage ” Draw t h e p i c t u r e o f a p o l y n o m i a l .Here a s example P=z ˆ13 − ∗ yf=expand ( z ˆ13 − ∗ zˆ7+zˆ4 − z ˆ3+5 ∗ zˆ2+z+3+2 ∗ i )u=( f+c o n j u g a t e ( f ) ) / 2v= − i ∗ ( f − u )p1=i m p l i c i t p l o t ( u==0,(x , − − − − References.
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