Find good alt expression for hyp term of bgmlv3

main.tex

@@ -571,17 +571,19 @@ bgmlv3_d_upperbound_exp_term = (
 
 # Verify the simplified forms of the terms that will be mentioned in text
 
-var("chbv",domain="real") # symbol to represent ch_1^\beta(v)
+var("chb1v chb2v",domain="real") # symbol to represent ch_1^\beta(v)
+var("psi phi", domain="real") # symbol to represent ch_1^\beta(v) and
+# ch_2^\beta(v)
 
 assert bgmlv3_d_upperbound_const_term == ( 
 	(
 		# keep hold of this alternative expression:
 		bgmlv3_d_upperbound_const_term_alt := (
-			chbv
+			phi
 			+ beta*q
 		)
 	)
-	.subs(chbv == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
+	.subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
 	.expand()
 )
 
@@ -590,13 +592,25 @@ assert bgmlv3_d_upperbound_exp_term == (
 		# Keep hold of this alternative expression:
 		bgmlv3_d_upperbound_exp_term_alt :=
 		(
-			R*chbv
+			R*phi
 			+ (C - q)^2/2
 			+ R*beta*q
 			- D*R
 		)/(r-R)
 	)
-	.subs(chbv == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
+	.subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
+	.expand()
+)
+
+assert bgmlv3_d_upperbound_exp_term == (
+	(
+		# Keep hold of this alternative expression:
+		bgmlv3_d_upperbound_exp_term_alt2 :=
+		(
+			(psi - q)^2/2/(r-R)
+		)
+	)
+	.subs(psi == v.twist(beta).ch[1]) # subs real val of ch_1^\beta(v)
 	.expand()
 )
 \end{sagesilent}
@@ -611,7 +625,10 @@ The linear term in $r$ is
 $\sage{bgmlv3_d_upperbound_linear_term}$.
 Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
 and can be written:
-$\frac{R\chern^{\beta}_2(F) + (C-q)^2/2 + R\beta q - DR}{r-R}$.
+\bgroup
+\def\psi{\chern_1^{\beta}(F)}
+$\sage{bgmlv3_d_upperbound_exp_term_alt2}$.
+\egroup
 In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
 $\chern^{\beta}_2(F) = 0$,
 so some of these expressions simplify, and in particular, the constant and