Fehrian Branches or: How I Learned to Reduce y = x^2 + 1 and Love Mathematica

Good morning everyone. Or I should say, good afternoon. Thank you for your joining us today. I will show you how to draw a graph of the complex valued functions. As you already know, to depict a pair of complex numbers, you know domain and the co domain will you need four dimension. This is why we hesitate even when we want to draw a graph. First I would like to ask you a question in a whole pane. Okay, so are you interested in the mathematical force dimension? This doesn't refer to any occult thing concepts, but to a four dimensional space distinct from four dimensional space time. A purely four dimensional space. I'd like to briefly reveal the history of the four dimensional visualization. First man who may be able to see the fourth dimension was Charles Hallard Hinton. And his name his invention Hinton's cube and hypercube as a tesseract. So his contribution is. Okay, maybe his popularization of higher dimensional figures using three methods by exam examining their shadows, their cross sections and their geometrical nets. This is very important things. And later after the. Okay, just here and when the 20th century begins, some geometricians, some mathematicians try to make plaster models for the educational use mainly for depicts the complex functions graph such as Weierstrauss pay function depicted. Okay, here in the. This is a. This is a models stored in Gettingen University in Germany now. So. And then I want to emphasize today is I'm in a spot right on this first mathematician on Howard Franklin Fair. His contribution is probably the first man who creates the cross section of the complex function with our three dimensional space. And he advocate for a more meaningful approach to teaching complex roots in high school mathematics. But okay, this is an important thing. He did not provide a precise mathematical definition for his cross section. He's just calling them real and complex branches. But the use of this terminology that lacks conventional accuracy in the fields of both complex analysis and algebraic geometry. However, this is today's point. I want to advocate say that with Urgham language his creation. You mean. Uram language means computer algebra and algebraic algorithms have the potential to substantiate phase application of the term branch. Okay, so let's start what are branches anyway? So the according to the properties and the behaviors of quadratics that we learned in middle school mathematics, the vertex of the parabola would be the lowest point on the graph if the quadratic were positive and opened upward. So on the contrary, when the highest point if the quadratic were negative and opened down, you see, it's okay, maybe. Okay, so we can combine these two graphs into one on purpose. On purpose to create a singular point at the origin, you know, in other words, tuck node at the origin. And the equation then becomes kartic function that is the product of two, you know, polynomials. So but this is no longer an irreducible polynomials. No. And another example is this one, a cubic function that also has a singularity at the origin, the ordinary double point or cool node at the origin. So this equation produces a graph like this. So this cannot be further factored as a polynomial, but using the formal power series. So we can make a. We can factor them. The function can be resolved into the product of the irreducible factors. And each factor is called a branch in a sense of algebraic geometry. In the example above, there are the two branches in this example. So back in 1945, as I said right now, Howard Franklin Fair put forward his real complex branches, mainly for educational use. So this picture is very impressive. And we can see each roots complex roots on the complex plane very easily. But his branch is not a branch used in the conventional traditional math fields. That's a problem. Yes. Okay, so from the point of view of the algebraic geometry, the quadratic equation shown in the previous slide can be factorized as this equation. So meaning, okay, that graphs left and right always branches. So branches is algebraic geometry and ones in complex analysis meet here the same meaning share the same meaning. Okay, so how to, how to justify Fayers branch? I just name it the Fair branches. We can do it with Mathematica. Okay, I'll show you step by step. So okay, here the example includes very basic quadratic equation x squared minus y + 1 equals 0, right? So replace all real variables X and Y in an equation and F with complex ones either replace all operators. So now take a cross section of the four dimensional space. So set V to zero, that is, restricts the equation to the three subspace X, Y and U from the X, Y, U v four dimensional space. So just cut the these three axes out of four to render the three dimensional cross sections of the fourth dimension. Now this, this is very basic operations of the Wuhan language. And okay, so reproduce, place the. Okay, we get the rule V equals to zero and then use equality of complex numbers. Means two complex numbers is equal if and only if this real part is zero and an imaginary part are there. So we can make this these operations. So we can reduce this formula into this one. Reduce expressions over reals. So the using the isomer satisfied by integral domains means A times B equals 0 equivalent to A equal A equals 0 or B equals 0. Right. So I use this axiom to solve the equation into more replace one step further. Okay, so, but fortunately reduce function of the WM language performs this equivalence transformation for us. This is very useful, very handy functions. So we can get two. Two terms joined by OR operator. This is very important things you have to look at. And with this, okay, here we can interpret or operator as timed operators multiplication like. Like a multiplication. The factorization. Okay, this is a logical factorization. And then we can solve this under determined simultaneous equations in X, Y and U. Okay, this Wolfram's language is a solve function is very, very powerful. It can solve under the. In the case of the equations are under determined. So solve is 2 parent 22 list list of list 2 list here 0y1 minus y square and x01 plus x square. This means, you know it. It looks like. It doesn't look like a factorization, but. But logically this is a logical factorization to generate fan branches. So now we ready to, you know, plot all branches. You know, they are parameterized perian branches like this. So using a normal parametric plot ID 3D function to see. Okay, in this case a 2 parabola like this form. So, okay, so let's consider defining a function user defined function called enumerate branches that contains all steps we have so far. Okay, the first enumerate branches and a render cross section user defined function. Okay, this we can define these two functions and show the branches for the cubic equation, for example, okay, as a application of this two user defined functions. So here this cubic function probably it will generalize the one real root and two complex roots. So I start the calculation. So it generates enumerate branches. Generate three branches here and every branch is in a curly braces. So we're in the cross section depicts the cross section. The entire cross section. Here you can see the real branch and colored in black and the convex branch and color in red. So we can apply this concept, you know, the first expand the polynomial into a complex domain and then reduce it and solve this under determined system equations. We can apply these steps for any planar cards like. Okay, this is the very simplest example is a circle, right? Okay, in this situation you can notice that the second argument of the enumerate branches is number two is equal to zero. That means we will take the subspace X U and V, not X Y X, Y U. You know, I will take X UB instead. So in this case, invert branches works well and it use four branches here and render cross section depicts all branches in the. In the cross section subspace here and then, okay, you can see the center. That's ordinary circle in gray color. But you know, in the complex domain two hyperbolas you can see. So when we expand any planar curve into a complex domain, so in this case a circle, we can interpret, okay, all these curves, you know, including a circle and the two parabola two hyperbolas as. Okay, this set of three curves are circle, complex, circle, complex five circle you. Okay, so far some issues, some problems. I realize some problems because this is a polynomial operations. So when the polynomials degree is greater than 4, so the reduce it doesn't work well. Okay. In this situation is a three leaf clover. This case it doesn't work unfortunately. Okay, the last my presentation is okay, elliptic curve catalog of elliptic curves shown on the Wikipedia page. You can see the original two dimensional chart on the Wikipedia page. We can extend those every two dimensional chart into a 3D subspace. That's a cross section. So let's start the calculation like this. So enumerate all branches and okay, so put them into three subspace to show the branches like these ones. Okay, so I noticed. Okay, gray. Gray curves are the ordinary planar curves. So let's. Okay, so you can see the whole charts in a graphics grid. Yeah, this is the end of my presentation. Thank you very much and I appreciate many people. It's a cooperation. Thank you very much. If you have any questions so you can put this secure API or later, please send me a mail. So in any ways, I welcome your questions every time after that. Thank you very much.