"What is proved about numbers will be a fact in any universe --Julia Robinson
"But will it be relevant?"--Barry Dayton

# Plane Curve Book The book is now out in Kindle, paperback and notebook formats from Wolfram Media. Go to wolfr.am/DaytonCurves for current information and links.

In addition there is a free summary of the book in Wolfram's The Mathematica Journal.

For the convenience of all users of the software defined in this book we have an index to Global Functions (PDF) which lists all the Global Functions (Appendix 2) alphabetically, giving syntax and location, by section, where further information on this function can be found in the book, chapter notebooks or Global Function notebook.

### Notes on Bezout's Theorem

(January 2019) I have written a new sub-section of Appendix 1 which explains Bezout's theorem in terms of the linear algebra developed in this Appendix. A PDF version (108KB) is here. The notebook version of this material is now included as section A.7 in the Appendix 1 notebook available in Chapter Notebooks.

### Execution Errors in Kindle and Print Versions

One challenge in production of this book is that both the typesetting an execution of code is done by Mathematica at the same time. For technical reasons some code did not execute properly when the the fixed print versions were typeset. These will run correctly in the notebook versions if properly initialized, you may need to re-run them from the downloaded files. Here are corrections:

• Section 1.6 page 16 (Make sure for ```In the cell property "evaluatable" is set.) In    gs = sqFree[g, x, y, 1.*^-12] Out   0.132808 x - 0.132808 x^3 - 0.132808 y + 0.132808 x^2 y - 0.132808 x y^2 + 0.132808 y^3 In    gs = Expand[gs/Coefficient[gs, x^3]] Out   -1. x + 1. x^3 + 1. y - 1. x^2 y + 1. x y^2 - 1. y^3 ```
• ```Section 3.2, page 45 (computer error) In    run2 = tangentRealPoints[f, k, x, y] Out   {{1.15237, 1.27611}, {0.707107, 0.5}, {0.707107, 0.5}, {0., -0.5}, {0., -0.5}, {0.928493, -0.857753}, {-0.754479, 0.417765}, {-0.707107, 0.5}, {-0.707107, 0.5}} ```
• ```Section 3.4, page 50 (failure to update from earlier version) In    fbow = x^4 - x^2 y + y^3;           fmin = -x^2 y + y^3;           infiniteRealPoints[fmin, x, y] Out   {{-5.22164, -5.22164, 0}, {-4.1098, 0., 0}, {-3.38833, 3.38833, 0}} ```
• ```Section 3.5,page 53 (failure to update) In    fm1 = Factor[pointMinForm[f, {-2, 0}, x, y]] Out   16 (x - y) (x + y) In    fm2 = Factor[pointMinForm[f, {2, 0}, x, y]] Out   16 (x - y) (x + y) ```
• ```Section 5.3, pages 98-99 (Make sure dTol = 1.*^-12) In    intersectionMultiplicity[y - x^2 + 1, y^2 - x^2 - 5, {2, 3}, dTol] Out   1 In    intersectionMultiplicity[y + 1/3 x^2 - 2 x - 1/3, y^2 - x^2 - 5, {2, 3}, dTol] Out   2 In    intersectionMultiplicity[y - x^3, y, {0, 0}, dTol] Out   3 In    intersectionMultiplicity[y - x - x^3, y - x - y^3, {0, 0}, dTol] Out   5 In    intersectionMultiplicity[y^2 - x^3, y - 3 x, {0, 0}, dTol] Out   2 In    intersectionMultiplicity[x^2 + y^2, y - 3 x, {0, 0}, dTol] Out   2 ```
• ```Section 7.6, p.167 (Failure to update from earlier version) In   {intersectionMultiplicity[y^2 - x^3 - x^2, l, {0, 0}, dTol],           intersectionMultiplicity[y^2 - x^3, l, {0, 0}, dTol],           intersectionMultiplicity[(y - x^2) (y + x^4), l, {0, 0}, dTol],           intersectionMultiplicity[x y (x - y), l, {0, 0}, dTol]} Out  {2, 2, 2, 3} Below this the variable d was assumed non-initialized but apparently was. The correction is In  Clear[d]          Expand[(d-1)(d-2)/2+2d-3] Out -2+d/2+d^2/2 On page 168 we have the same problem. Hopefully now d is unevaluated so In   Expand[2((d-1)(d-2)/2)+2d-3] Out -1-d+d^2   But this is d(d-1)-1. ```
• ```Section 8.5 p.194-195 The graphic for Example 3 was omitted. It does appear on p. 206. For original notebook version this is also a problem, the picture is in Section 8.7. ```
• ```Section 8.7 p. 204 Example A2 The graphic for Example A2, affine hyperbola, is wrong, the correct picture was shown on p 194. This can be corrected in the notebook version by changing the first line of in to           Hp = Table[{Sqrt[y^2 + 4], y}, {y, -6, 6, .201}] This avoids hitting the x-axis which is ambiguous. Let me know at barry@barryhdayton.us if you find other places where there is a problem. ```